:: Equations in Many Sorted Algebras
:: by Artur Korni{\l}owicz
::
:: Received May 30, 1997
:: Copyright (c) 1997 Association of Mizar Users
theorem Th1: :: EQUATION:1
theorem :: EQUATION:2
theorem Th3: :: EQUATION:3
theorem :: EQUATION:4
canceled;
theorem :: EQUATION:5
theorem Th6: :: EQUATION:6
theorem Th7: :: EQUATION:7
:: deftheorem Def1 defines "" EQUATION:def 1 :
theorem :: EQUATION:8
theorem :: EQUATION:9
theorem :: EQUATION:10
theorem :: EQUATION:11
theorem :: EQUATION:12
theorem :: EQUATION:13
theorem Th14: :: EQUATION:14
theorem Th15: :: EQUATION:15
theorem :: EQUATION:16
theorem :: EQUATION:17
theorem :: EQUATION:18
canceled;
theorem Th19: :: EQUATION:19
theorem Th20: :: EQUATION:20
theorem Th21: :: EQUATION:21
theorem Th22: :: EQUATION:22
:: deftheorem Def2 defines SuperAlgebraSet EQUATION:def 2 :
theorem Th23: :: EQUATION:23
theorem Th24: :: EQUATION:24
theorem :: EQUATION:25
theorem Th26: :: EQUATION:26
theorem Th27: :: EQUATION:27
theorem Th28: :: EQUATION:28
theorem :: EQUATION:29
theorem :: EQUATION:30
:: deftheorem defines TermAlg EQUATION:def 3 :
:: deftheorem defines Equations EQUATION:def 4 :
theorem Th31: :: EQUATION:31
theorem Th32: :: EQUATION:32
:: deftheorem Def5 defines |= EQUATION:def 5 :
:: deftheorem Def6 defines |= EQUATION:def 6 :
theorem Th33: :: EQUATION:33
theorem :: EQUATION:34
theorem Th35: :: EQUATION:35
theorem :: EQUATION:36
theorem Th37: :: EQUATION:37
theorem :: EQUATION:38
Lm1:
for S being non empty non void ManySortedSign
for s being SortSymbol of S
for e being Element of (Equations S) . s
for I being non empty set
for A being MSAlgebra-Family of I,S st ( for i being Element of I holds A . i |= e ) holds
product A |= e
theorem Th39: :: EQUATION:39
theorem :: EQUATION:40