let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of SCM st p c= s1 & p c= s2 holds
for i being Element of NAT
for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput ((ProgramPart s1),s1,i)) . da) * ((Comput ((ProgramPart s1),s1,i)) . db) = ((Comput ((ProgramPart s2),s2,i)) . da) * ((Comput ((ProgramPart s2),s2,i)) . db)
let s1, s2 be State of SCM; :: thesis: ( p c= s1 & p c= s2 implies for i being Element of NAT
for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput ((ProgramPart s1),s1,i)) . da) * ((Comput ((ProgramPart s1),s1,i)) . db) = ((Comput ((ProgramPart s2),s2,i)) . da) * ((Comput ((ProgramPart s2),s2,i)) . db) )
assume A1: ( p c= s1 & p c= s2 ) ; :: thesis: for i being Element of NAT
for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput ((ProgramPart s1),s1,i)) . da) * ((Comput ((ProgramPart s1),s1,i)) . db) = ((Comput ((ProgramPart s2),s2,i)) . da) * ((Comput ((ProgramPart s2),s2,i)) . db)
let i be Element of NAT ; :: thesis: for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput ((ProgramPart s1),s1,i)) . da) * ((Comput ((ProgramPart s1),s1,i)) . db) = ((Comput ((ProgramPart s2),s2,i)) . da) * ((Comput ((ProgramPart s2),s2,i)) . db)
let da, db be Data-Location ; :: thesis: for I being Instruction of SCM st I = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) & I = MultBy (da,db) & da in dom p holds
((Comput ((ProgramPart s1),s1,i)) . da) * ((Comput ((ProgramPart s1),s1,i)) . db) = ((Comput ((ProgramPart s2),s2,i)) . da) * ((Comput ((ProgramPart s2),s2,i)) . db)
let I be Instruction of SCM; :: thesis: ( I = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) & I = MultBy (da,db) & da in dom p implies ((Comput ((ProgramPart s1),s1,i)) . da) * ((Comput ((ProgramPart s1),s1,i)) . db) = ((Comput ((ProgramPart s2),s2,i)) . da) * ((Comput ((ProgramPart s2),s2,i)) . db) )
assume A2: I = CurInstr ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) ; :: thesis: ( not I = MultBy (da,db) or not da in dom p or ((Comput ((ProgramPart s1),s1,i)) . da) * ((Comput ((ProgramPart s1),s1,i)) . db) = ((Comput ((ProgramPart s2),s2,i)) . da) * ((Comput ((ProgramPart s2),s2,i)) . db) )
set Cs2i1 = Comput ((ProgramPart s2),s2,(i + 1));
set Cs2i = Comput ((ProgramPart s2),s2,i);
T: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,i)) by AMI_1:123;
A3: Comput ((ProgramPart s2),s2,(i + 1)) = Following ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,i))),(Comput ((ProgramPart s2),s2,i)))),(Comput ((ProgramPart s2),s2,i))) by T ;
assume that
A4: I = MultBy (da,db) and
A5: ( da in dom p & ((Comput ((ProgramPart s1),s1,i)) . da) * ((Comput ((ProgramPart s1),s1,i)) . db) <> ((Comput ((ProgramPart s2),s2,i)) . da) * ((Comput ((ProgramPart s2),s2,i)) . db) ) ; :: thesis: contradiction
TY: ProgramPart s2 = ProgramPart (Comput ((ProgramPart s2),s2,i)) by AMI_1:123;
I = CurInstr ((ProgramPart s2),(Comput ((ProgramPart s2),s2,i))) by A1, A2, Th87;
then A6: (Comput ((ProgramPart s2),s2,(i + 1))) . da = ((Comput ((ProgramPart s2),s2,i)) . da) * ((Comput ((ProgramPart s2),s2,i)) . db) by A3, A4, TY, AMI_3:11;
set Cs1i1 = Comput ((ProgramPart s1),s1,(i + 1));
set Cs1i = Comput ((ProgramPart s1),s1,i);
A7: ( da in dom p implies ( ((Comput ((ProgramPart s1),s1,(i + 1))) | (dom p)) . da = (Comput ((ProgramPart s1),s1,(i + 1))) . da & ((Comput ((ProgramPart s2),s2,(i + 1))) | (dom p)) . da = (Comput ((ProgramPart s2),s2,(i + 1))) . da ) ) by FUNCT_1:72;
T: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,i)) by AMI_1:123;
TX: ProgramPart s1 = ProgramPart (Comput ((ProgramPart s1),s1,i)) by AMI_1:123;
Comput ((ProgramPart s1),s1,(i + 1)) = Following ((ProgramPart s1),(Comput ((ProgramPart s1),s1,i))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,i))),(Comput ((ProgramPart s1),s1,i)))),(Comput ((ProgramPart s1),s1,i))) by T ;
then (Comput ((ProgramPart s1),s1,(i + 1))) . da = ((Comput ((ProgramPart s1),s1,i)) . da) * ((Comput ((ProgramPart s1),s1,i)) . db) by A2, A4, TX, AMI_3:11;
hence contradiction by A1, A7, A5, A6, EXTPRO_1:def 9; :: thesis: verum