let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of st p c= s1 & p c= s2 holds
for i being Element of NAT
for k1, k2 being Integer
for a, b being Int_position st CurInstr (Computation s1,i) = MultBy a,k1,b,k2 & a in dom p & DataLoc ((Computation s1,i) . a),k1 in dom p holds
((Computation s1,i) . (DataLoc ((Computation s1,i) . a),k1)) * ((Computation s1,i) . (DataLoc ((Computation s1,i) . b),k2)) = ((Computation s2,i) . (DataLoc ((Computation s2,i) . a),k1)) * ((Computation s2,i) . (DataLoc ((Computation s2,i) . b),k2))
let s1, s2 be State of ; :: thesis: ( p c= s1 & p c= s2 implies for i being Element of NAT
for k1, k2 being Integer
for a, b being Int_position st CurInstr (Computation s1,i) = MultBy a,k1,b,k2 & a in dom p & DataLoc ((Computation s1,i) . a),k1 in dom p holds
((Computation s1,i) . (DataLoc ((Computation s1,i) . a),k1)) * ((Computation s1,i) . (DataLoc ((Computation s1,i) . b),k2)) = ((Computation s2,i) . (DataLoc ((Computation s2,i) . a),k1)) * ((Computation s2,i) . (DataLoc ((Computation s2,i) . b),k2)) )
assume A1: ( p c= s1 & p c= s2 ) ; :: thesis: for i being Element of NAT
for k1, k2 being Integer
for a, b being Int_position st CurInstr (Computation s1,i) = MultBy a,k1,b,k2 & a in dom p & DataLoc ((Computation s1,i) . a),k1 in dom p holds
((Computation s1,i) . (DataLoc ((Computation s1,i) . a),k1)) * ((Computation s1,i) . (DataLoc ((Computation s1,i) . b),k2)) = ((Computation s2,i) . (DataLoc ((Computation s2,i) . a),k1)) * ((Computation s2,i) . (DataLoc ((Computation s2,i) . b),k2))
let i be Element of NAT ; :: thesis: for k1, k2 being Integer
for a, b being Int_position st CurInstr (Computation s1,i) = MultBy a,k1,b,k2 & a in dom p & DataLoc ((Computation s1,i) . a),k1 in dom p holds
((Computation s1,i) . (DataLoc ((Computation s1,i) . a),k1)) * ((Computation s1,i) . (DataLoc ((Computation s1,i) . b),k2)) = ((Computation s2,i) . (DataLoc ((Computation s2,i) . a),k1)) * ((Computation s2,i) . (DataLoc ((Computation s2,i) . b),k2))
let k1, k2 be Integer; :: thesis: for a, b being Int_position st CurInstr (Computation s1,i) = MultBy a,k1,b,k2 & a in dom p & DataLoc ((Computation s1,i) . a),k1 in dom p holds
((Computation s1,i) . (DataLoc ((Computation s1,i) . a),k1)) * ((Computation s1,i) . (DataLoc ((Computation s1,i) . b),k2)) = ((Computation s2,i) . (DataLoc ((Computation s2,i) . a),k1)) * ((Computation s2,i) . (DataLoc ((Computation s2,i) . b),k2))
let a, b be Int_position ; :: thesis: ( CurInstr (Computation s1,i) = MultBy a,k1,b,k2 & a in dom p & DataLoc ((Computation s1,i) . a),k1 in dom p implies ((Computation s1,i) . (DataLoc ((Computation s1,i) . a),k1)) * ((Computation s1,i) . (DataLoc ((Computation s1,i) . b),k2)) = ((Computation s2,i) . (DataLoc ((Computation s2,i) . a),k1)) * ((Computation s2,i) . (DataLoc ((Computation s2,i) . b),k2)) )
set I = CurInstr (Computation s1,i);
set Cs1i = Computation s1,i;
set Cs2i = Computation s2,i;
assume that
A2: CurInstr (Computation s1,i) = MultBy a,k1,b,k2 and
A3: a in dom p and
A4: DataLoc ((Computation s1,i) . a),k1 in dom p ; :: thesis: ((Computation s1,i) . (DataLoc ((Computation s1,i) . a),k1)) * ((Computation s1,i) . (DataLoc ((Computation s1,i) . b),k2)) = ((Computation s2,i) . (DataLoc ((Computation s2,i) . a),k1)) * ((Computation s2,i) . (DataLoc ((Computation s2,i) . b),k2))
( a in dom p implies ( ((Computation s1,i) | (dom p)) . a = (Computation s1,i) . a & ((Computation s2,i) | (dom p)) . a = (Computation s2,i) . a ) ) by FUNCT_1:72;
then A5: (Computation s1,i) . a = (Computation s2,i) . a by A1, A3, AMI_1:def 25;
set Cs2i1 = Computation s2,(i + 1);
set Cs1i1 = Computation s1,(i + 1);
set D11 = (Computation s1,(i + 1)) . (DataLoc ((Computation s1,i) . a),k1);
set D21 = (Computation s2,(i + 1)) . (DataLoc ((Computation s2,i) . a),k1);
Computation s1,(i + 1) = Following (Computation s1,i) by AMI_1:14
.= Exec (CurInstr (Computation s1,i)),(Computation s1,i) ;
then A6: (Computation s1,(i + 1)) . (DataLoc ((Computation s1,i) . a),k1) = ((Computation s1,i) . (DataLoc ((Computation s1,i) . a),k1)) * ((Computation s1,i) . (DataLoc ((Computation s1,i) . b),k2)) by A2, SCMPDS_2:63;
A7: Computation s2,(i + 1) = Following (Computation s2,i) by AMI_1:14
.= Exec (CurInstr (Computation s2,i)),(Computation s2,i) ;
A8: ( DataLoc ((Computation s1,i) . a),k1 in dom p implies ( ((Computation s1,(i + 1)) | (dom p)) . (DataLoc ((Computation s1,i) . a),k1) = (Computation s1,(i + 1)) . (DataLoc ((Computation s1,i) . a),k1) & ((Computation s2,(i + 1)) | (dom p)) . (DataLoc ((Computation s1,i) . a),k1) = (Computation s2,(i + 1)) . (DataLoc ((Computation s1,i) . a),k1) ) ) by FUNCT_1:72;
CurInstr (Computation s1,i) = CurInstr (Computation s2,i) by A1, Th23;
then (Computation s2,(i + 1)) . (DataLoc ((Computation s2,i) . a),k1) = ((Computation s2,i) . (DataLoc ((Computation s2,i) . a),k1)) * ((Computation s2,i) . (DataLoc ((Computation s2,i) . b),k2)) by A7, A2, SCMPDS_2:63;
hence ((Computation s1,i) . (DataLoc ((Computation s1,i) . a),k1)) * ((Computation s1,i) . (DataLoc ((Computation s1,i) . b),k2)) = ((Computation s2,i) . (DataLoc ((Computation s2,i) . a),k1)) * ((Computation s2,i) . (DataLoc ((Computation s2,i) . b),k2)) by A1, A8, A4, A5, A6, AMI_1:def 25; :: thesis: verum