:: FVSUM_1 semantic presentation
theorem Th1: :: FVSUM_1:1
canceled;
theorem Th2: :: FVSUM_1:2
theorem Th3: :: FVSUM_1:3
theorem Th4: :: FVSUM_1:4
theorem Th5: :: FVSUM_1:5
canceled;
theorem Th6: :: FVSUM_1:6
theorem Th7: :: FVSUM_1:7
theorem Th8: :: FVSUM_1:8
theorem Th9: :: FVSUM_1:9
theorem Th10: :: FVSUM_1:10
theorem Th11: :: FVSUM_1:11
theorem Th12: :: FVSUM_1:12
:: deftheorem Def1 defines multfield FVSUM_1:def 1 :
:: deftheorem Def2 defines diffield FVSUM_1:def 2 :
theorem Th13: :: FVSUM_1:13
canceled;
theorem Th14: :: FVSUM_1:14
Lemma46:
for K being non empty HGrStr
for a, b being Element of K holds (the mult of K [;] b,(id the carrier of K)) . a = b * a
theorem Th15: :: FVSUM_1:15
theorem Th16: :: FVSUM_1:16
theorem Th17: :: FVSUM_1:17
theorem Th18: :: FVSUM_1:18
theorem Th19: :: FVSUM_1:19
:: deftheorem Def3 defines + FVSUM_1:def 3 :
theorem Th20: :: FVSUM_1:20
canceled;
theorem Th21: :: FVSUM_1:21
theorem Th22: :: FVSUM_1:22
theorem Th23: :: FVSUM_1:23
theorem Th24: :: FVSUM_1:24
theorem Th25: :: FVSUM_1:25
theorem Th26: :: FVSUM_1:26
theorem Th27: :: FVSUM_1:27
Lemma62:
for i being Element of NAT
for K being non empty right_zeroed left_zeroed LoopStr
for R being Element of i -tuples_on the carrier of K holds R + (i |-> (0. K)) = R
theorem Th28: :: FVSUM_1:28
:: deftheorem Def4 defines - FVSUM_1:def 4 :
theorem Th29: :: FVSUM_1:29
canceled;
theorem Th30: :: FVSUM_1:30
theorem Th31: :: FVSUM_1:31
theorem Th32: :: FVSUM_1:32
theorem Th33: :: FVSUM_1:33
theorem Th34: :: FVSUM_1:34
Lemma67:
for i being Element of NAT
for K being non empty add-associative right_zeroed right_complementable left_zeroed LoopStr
for R being Element of i -tuples_on the carrier of K holds R + (- R) = i |-> (0. K)
theorem Th35: :: FVSUM_1:35
theorem Th36: :: FVSUM_1:36
theorem Th37: :: FVSUM_1:37
theorem Th38: :: FVSUM_1:38
Lemma71:
for i being Element of NAT
for K being non empty add-associative right_zeroed right_complementable left_zeroed LoopStr
for R1, R, R2 being Element of i -tuples_on the carrier of K st R1 + R = R2 + R holds
R1 = R2
theorem Th39: :: FVSUM_1:39
theorem Th40: :: FVSUM_1:40
:: deftheorem Def5 defines - FVSUM_1:def 5 :
theorem Th41: :: FVSUM_1:41
canceled;
theorem Th42: :: FVSUM_1:42
theorem Th43: :: FVSUM_1:43
theorem Th44: :: FVSUM_1:44
theorem Th45: :: FVSUM_1:45
theorem Th46: :: FVSUM_1:46
theorem Th47: :: FVSUM_1:47
theorem Th48: :: FVSUM_1:48
theorem Th49: :: FVSUM_1:49
theorem Th50: :: FVSUM_1:50
theorem Th51: :: FVSUM_1:51
theorem Th52: :: FVSUM_1:52
theorem Th53: :: FVSUM_1:53
theorem Th54: :: FVSUM_1:54
theorem Th55: :: FVSUM_1:55
theorem Th56: :: FVSUM_1:56
theorem Th57: :: FVSUM_1:57
theorem Th58: :: FVSUM_1:58
theorem Th59: :: FVSUM_1:59
theorem Th60: :: FVSUM_1:60
theorem Th61: :: FVSUM_1:61
:: deftheorem Def6 defines * FVSUM_1:def 6 :
theorem Th62: :: FVSUM_1:62
theorem Th63: :: FVSUM_1:63
theorem Th64: :: FVSUM_1:64
theorem Th65: :: FVSUM_1:65
theorem Th66: :: FVSUM_1:66
theorem Th67: :: FVSUM_1:67
theorem Th68: :: FVSUM_1:68
theorem Th69: :: FVSUM_1:69
theorem Th70: :: FVSUM_1:70
theorem Th71: :: FVSUM_1:71
theorem Th72: :: FVSUM_1:72
:: deftheorem Def7 defines mlt FVSUM_1:def 7 :
theorem Th73: :: FVSUM_1:73
theorem Th74: :: FVSUM_1:74
theorem Th75: :: FVSUM_1:75
theorem Th76: :: FVSUM_1:76
Lemma92:
for i being Element of NAT
for K being non empty HGrStr
for a1, a2 being Element of K
for R1, R2 being Element of i -tuples_on the carrier of K holds mlt (R1 ^ <*a1*>),(R2 ^ <*a2*>) = (mlt R1,R2) ^ <*(a1 * a2)*>
Lemma93:
for K being non empty HGrStr
for a1, a2, b1, b2 being Element of K holds mlt <*a1,a2*>,<*b1,b2*> = <*(a1 * b1),(a2 * b2)*>
theorem Th77: :: FVSUM_1:77
theorem Th78: :: FVSUM_1:78
theorem Th79: :: FVSUM_1:79
theorem Th80: :: FVSUM_1:80
theorem Th81: :: FVSUM_1:81
theorem Th82: :: FVSUM_1:82
theorem Th83: :: FVSUM_1:83
theorem Th84: :: FVSUM_1:84
:: deftheorem Def8 defines Sum FVSUM_1:def 8 :
theorem Th85: :: FVSUM_1:85
canceled;
theorem Th86: :: FVSUM_1:86
canceled;
theorem Th87: :: FVSUM_1:87
theorem Th88: :: FVSUM_1:88
canceled;
theorem Th89: :: FVSUM_1:89
theorem Th90: :: FVSUM_1:90
canceled;
theorem Th91: :: FVSUM_1:91
canceled;
theorem Th92: :: FVSUM_1:92
theorem Th93: :: FVSUM_1:93
theorem Th94: :: FVSUM_1:94
theorem Th95: :: FVSUM_1:95
theorem Th96: :: FVSUM_1:96
definition
canceled;
end;
:: deftheorem Def9 FVSUM_1:def 9 :
canceled;
theorem Th97: :: FVSUM_1:97
canceled;
theorem Th98: :: FVSUM_1:98
theorem Th99: :: FVSUM_1:99
theorem Th100: :: FVSUM_1:100
theorem Th101: :: FVSUM_1:101
theorem Th102: :: FVSUM_1:102
theorem Th103: :: FVSUM_1:103
theorem Th104: :: FVSUM_1:104
theorem Th105: :: FVSUM_1:105
theorem Th106: :: FVSUM_1:106
theorem Th107: :: FVSUM_1:107
theorem Th108: :: FVSUM_1:108
theorem Th109: :: FVSUM_1:109
theorem Th110: :: FVSUM_1:110
theorem Th111: :: FVSUM_1:111
theorem Th112: :: FVSUM_1:112
:: deftheorem Def10 defines "*" FVSUM_1:def 10 :
theorem Th113: :: FVSUM_1:113
theorem Th114: :: FVSUM_1:114
theorem Th115: :: FVSUM_1:115