let k be Element of NAT ; :: thesis:
let q be NAT -defined the Instructions of SCM -valued finite non halt-free Function; :: thesis:
let p be non empty q -autonomic FinPartState of SCM; :: thesis:
let s1, s2 be State of SCM; :: thesis:
assume that
A1: IC in dom p and
A2: p c= s1 and
A3: IncIC (p,k) c= s2 ; :: thesis:
B1: IC in dom p by A1;
B2: p c= s1 by A2;
let P1, P2 be Instruction-Sequence of SCM; :: thesis:
assume A4: ( q c= P1 & Reloc (q,k) c= P2 ) ; :: thesis:
A5: Reloc (q,k) c= P2 by A4;
A8: q c= P1 by A4;
set s3 = s1 +* (DataPart s2);
defpred S₁[ Element of NAT ] means ( (IC (Comput (P1,s1,$1))) + k = IC (Comput (P2,s2,$1)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,$1)))),k) = CurInstr (P2,(Comput (P2,s2,$1))) & (Comput (P1,s1,$1)) | (dom (DataPart p)) = (Comput (P2,s2,$1)) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),$1)) = DataPart (Comput (P2,s2,$1)) );
A10: p c= s1 +* (DataPart s2) by A2, A3, MEMSTR_0:61;
A126: for i being Element of NAT st S₁[i] holds
S₁[i + 1]

set DPp = DataPart p;
let i be Element of NAT ; :: thesis:
assume that
A11: (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) and
A12: IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) and
A13: (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) and
A14: DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) ; :: thesis:
set Cs2i1 = Comput (P2,s2,(i + 1));
set Cs3i = Comput (P1,(s1 +* (DataPart s2)),i);
set Cs2i = Comput (P2,s2,i);
dom (Comput (P2,s2,(i + 1))) = the carrier of SCM by PARTFUN1:def 2;
then A15: dom (Comput (P2,s2,(i + 1))) = {(IC )} \/ (Data-Locations ) by STRUCT_0:4;
set Cs3i1 = Comput (P1,(s1 +* (DataPart s2)),(i + 1));
A16: dom (DataPart (Comput (P2,s2,i))) = Data-Locations by MEMSTR_0:9;
A17: dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) = Data-Locations by MEMSTR_0:9;
A18: dom (DataPart (Comput (P2,s2,(i + 1)))) = Data-Locations by MEMSTR_0:9;

A19: now

let x be set ; :: thesis:
assume that
A20: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) and
A21: (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . x = (Comput (P2,s2,(i + 1))) . x ; :: thesis:
thus (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (Comput (P2,s2,(i + 1))) . x by A20, A21, FUNCT_1:47
.= (DataPart (Comput (P2,s2,(i + 1)))) . x by A17, A18, A20, FUNCT_1:47 ; :: thesis:

end;

A22: dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),i))) = Data-Locations by MEMSTR_0:9;

A23: now

let x be set ; :: thesis:
assume that
A24: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) and
A25: ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . x = (Comput (P1,(s1 +* (DataPart s2)),i)) . x & (Comput (P2,s2,(i + 1))) . x = (Comput (P2,s2,i)) . x ) ; :: thesis:
(DataPart (Comput (P1,(s1 +* (DataPart s2)),i))) . x = (Comput (P1,(s1 +* (DataPart s2)),i)) . x by A22, A17, A24, FUNCT_1:47;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A14, A16, A17, A19, A24, A25, FUNCT_1:47; :: thesis:

end;

A26: now

let s be State of SCM; :: thesis:
let d be Data-Location ; :: thesis:
d in Data-Locations by AMI_3:27, AMI_3:def 2;
hence d in dom (DataPart s) by MEMSTR_0:9; :: thesis:

end;

A27: now

let d be Data-Location ; :: thesis:
A28: d in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),i))) by A26;
hence (Comput (P1,(s1 +* (DataPart s2)),i)) . d = (DataPart (Comput (P1,(s1 +* (DataPart s2)),i))) . d by FUNCT_1:47
.= (Comput (P2,s2,i)) . d by A14, A28, FUNCT_1:47 ;
:: thesis:

end;

set Cs1i1 = Comput (P1,s1,(i + 1));
set Cs1i = Comput (P1,s1,i);
dom (Comput (P1,s1,(i + 1))) = the carrier of SCM by PARTFUN1:def 2;
then A29: dom (Comput (P1,s1,(i + 1))) = {(IC )} \/ (Data-Locations ) by STRUCT_0:4;
dom (DataPart p) = (dom p) /\ (Data-Locations ) by RELAT_1:61;
then A30: dom (DataPart p) c= {(IC )} \/ (Data-Locations ) by XBOOLE_1:10, XBOOLE_1:17;
A31: dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) = (dom (Comput (P1,s1,(i + 1)))) /\ (dom (DataPart p)) by RELAT_1:61
.= dom (DataPart p) by A29, A30, XBOOLE_1:28 ;

A32: now

reconsider loc = IC (Comput (P1,s1,(i + 1))) as Element of NAT ;
assume A33: (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) ; :: thesis:
reconsider kk = loc as Element of NAT ;
A34: loc in dom q by A8, B2, AMISTD_5:def 4;
A35: loc + k in dom (Reloc (q,k)) by A34, COMPOS_1:46;
A36: dom P2 = NAT by PARTFUN1:def 2;
dom P1 = NAT by PARTFUN1:def 2;
then CurInstr (P1,(Comput (P1,s1,(i + 1)))) = P1 . loc by PARTFUN1:def 6
.= q . loc by A34, A4, GRFUNC_1:2
.= q . loc ;
hence IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = (Reloc (q,k)) . (loc + k) by A34, COMPOS_1:35
.= P2 . (IC (Comput (P2,s2,(i + 1)))) by A33, A35, A5, GRFUNC_1:2
.= CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A36, PARTFUN1:def 6 ;
:: thesis:

end;

set I = CurInstr (P1,(Comput (P1,s1,i)));
A37: Comput (P2,s2,(i + 1)) = Following (P2,(Comput (P2,s2,i))) by EXTPRO_1:3
.= Exec ((CurInstr (P2,(Comput (P2,s2,i)))),(Comput (P2,s2,i))) ;
dom (Comput (P2,s2,i)) = the carrier of SCM by PARTFUN1:def 2;
then A38: dom (Comput (P2,s2,i)) = {(IC )} \/ (Data-Locations ) by STRUCT_0:4;
dom (Comput (P1,s1,i)) = the carrier of SCM by PARTFUN1:def 2;
then A39: dom (Comput (P1,s1,i)) = {(IC )} \/ (Data-Locations ) by STRUCT_0:4;
A40: dom ((Comput (P1,s1,i)) | (dom (DataPart p))) = (dom (Comput (P1,s1,i))) /\ (dom (DataPart p)) by RELAT_1:61
.= dom (DataPart p) by A39, A30, XBOOLE_1:28 ;
A41: Comput (P1,(s1 +* (DataPart s2)),(i + 1)) = Following (P1,(Comput (P1,(s1 +* (DataPart s2)),i))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,(s1 +* (DataPart s2)),i))) by A2, A10, A4, AMISTD_5:7 ;
A43: dom ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) = (dom (Comput (P2,s2,(i + 1)))) /\ (dom (DataPart p)) by RELAT_1:61
.= dom (DataPart p) by A15, A30, XBOOLE_1:28 ;

A44: now

let x be set ; :: thesis:
let d be Data-Location ; :: thesis:
assume that
A45: ( d = x & d in dom (DataPart p) ) and
A46: (Comput (P1,s1,(i + 1))) . d = (Comput (P2,s2,(i + 1))) . d ; :: thesis:
thus ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = (Comput (P2,s2,(i + 1))) . d by A31, A45, A46, FUNCT_1:47
.= ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A43, A45, FUNCT_1:47 ; :: thesis:

end;

A47: dom ((Comput (P2,s2,i)) | (dom (DataPart p))) = (dom (Comput (P2,s2,i))) /\ (dom (DataPart p)) by RELAT_1:61
.= dom (DataPart p) by A38, A30, XBOOLE_1:28 ;

A48: now

let x be set ; :: thesis:
let d be Data-Location ; :: thesis:
assume that
A49: d = x and
A50: d in dom (DataPart p) and
A51: ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) ; :: thesis:
( ((Comput (P1,s1,i)) | (dom (DataPart p))) . d = (Comput (P1,s1,i)) . d & ((Comput (P2,s2,i)) | (dom (DataPart p))) . d = (Comput (P2,s2,i)) . d ) by A40, A47, A50, FUNCT_1:47;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = (Comput (P2,s2,(i + 1))) . d by A13, A31, A49, A50, A51, FUNCT_1:47
.= ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A43, A49, A50, FUNCT_1:47 ;
:: thesis:

end;

reconsider j = IC (Comput (P1,s1,i)) as Element of NAT ;
A52: Comput (P1,s1,(i + 1)) = Following (P1,(Comput (P1,s1,i))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i))) ;
A53: succ ((IC (Comput (P1,s1,i))) + k) = (j + k) + 1 by NAT_1:38
.= (j + 1) + k
.= (succ j) + k by NAT_1:38 ;

per cases by AMI_5:5, NAT_1:32;

suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 0 ; :: thesis:

then A54: CurInstr (P1,(Comput (P1,s1,i))) = halt SCM by AMI_5:7;
thus (IC (Comput (P1,s1,(i + 1)))) + k = (IC (Comput (P1,s1,i))) + k by A52, A54, EXTPRO_1:def 3
.= IC (Comput (P2,s2,(i + 1))) by A11, A37, A54, A12, EXTPRO_1:def 3 ; :: thesis:
hence IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A32; :: thesis:
A55: Comput (P2,s2,(i + 1)) = Comput (P2,s2,i) by A37, A54, A12, EXTPRO_1:def 3;
hence (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A13, A52, A54, EXTPRO_1:def 3; :: thesis:
thus DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) by A14, A41, A54, A55, EXTPRO_1:def 3; :: thesis:

end;

suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 1 ; :: thesis:

then consider da, db being Data-Location such that
A56: CurInstr (P1,(Comput (P1,s1,i))) = da := db by AMI_5:8;
A57: IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = da := db by A56, COMPOS_1:11;
A58: (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . (IC ) = succ (IC (Comput (P1,s1,i))) by A56, AMI_3:2;
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A11, A12, A52, A37, A53, A57, AMI_3:2; :: thesis:
thus IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A11, A12, A32, A52, A37, A53, A57, A58, AMI_3:2; :: thesis:
A59: (Comput (P1,(s1 +* (DataPart s2)),i)) . db = (Comput (P2,s2,i)) . db by A27;

now

DataPart p c= p by RELAT_1:59;
then A60: dom (DataPart p) c= dom p by GRFUNC_1:2;
let x be set ; :: thesis:
assume A61: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis:
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then x in Data-Locations by A31, A61;
then reconsider d = x as Data-Location by AMI_3:27, AMI_3:def 2;

per cases ;

suppose A62: da = d ; :: thesis:

then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . db & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . db ) by A12, A52, A37, A56, A57, AMI_3:2;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A2, A10, A31, A44, A56, A59, A61, A60, A62, A4, AMI_5:17; :: thesis:

end;

suppose da <> d ; :: thesis:

then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A12, A52, A37, A56, A57, AMI_3:2;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A31, A48, A61; :: thesis:

end;

then (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) c= (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A31, A43, GRFUNC_1:2;
hence (Comput (P1,s1,(i + 1))) | (dom (DataPart p)) = (Comput (P2,s2,(i + 1))) | (dom (DataPart p)) by A31, A43, GRFUNC_1:3; :: thesis:

now

let x be set ; :: thesis:
assume A63: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis:
then reconsider d = x as Data-Location by A17, AMI_3:27, AMI_3:def 2;

per cases ;

suppose da = d ; :: thesis:

then ( (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . db & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . db ) by A12, A37, A41, A56, A57, AMI_3:2;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A27, A19, A63; :: thesis:

end;

suppose da <> d ; :: thesis:

then ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A12, A37, A41, A56, A57, AMI_3:2;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A23, A63; :: thesis:

end;

then DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) c= DataPart (Comput (P2,s2,(i + 1))) by A17, A18, GRFUNC_1:2;
hence DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) = DataPart (Comput (P2,s2,(i + 1))) by A17, A18, GRFUNC_1:3; :: thesis:

end;

suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 2 ; :: thesis:

then consider da, db being Data-Location such that
A64: CurInstr (P1,(Comput (P1,s1,i))) = AddTo (da,db) by AMI_5:9;
A65: IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = AddTo (da,db) by A64, COMPOS_1:11;
A66: (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . (IC ) = succ (IC (Comput (P1,s1,i))) by A64, AMI_3:3;
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A11, A12, A52, A37, A53, A65, AMI_3:3; :: thesis:
thus IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A11, A12, A32, A52, A37, A53, A65, A66, AMI_3:3; :: thesis:
A67: ( (Comput (P1,(s1 +* (DataPart s2)),i)) . da = (Comput (P2,s2,i)) . da & (Comput (P1,(s1 +* (DataPart s2)),i)) . db = (Comput (P2,s2,i)) . db ) by A27;

now

DataPart p c= p by RELAT_1:59;
then A68: dom (DataPart p) c= dom p by GRFUNC_1:2;
let x be set ; :: thesis:
assume A69: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis:
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then x in Data-Locations by A31, A69;
then reconsider d = x as Data-Location by AMI_3:27, AMI_3:def 2;

per cases ;

suppose A70: da = d ; :: thesis:

then ( (Comput (P1,s1,(i + 1))) . d = ((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) & (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db) ) by A12, A52, A37, A64, A65, AMI_3:3;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A2, A10, A31, A44, A64, A67, A69, A68, A70, A4, AMI_5:18; :: thesis:

end;

suppose da <> d ; :: thesis:

then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A12, A52, A37, A64, A65, AMI_3:3;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A31, A48, A69; :: thesis:

end;

now

let x be set ; :: thesis:
assume A71: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis:
then reconsider d = x as Data-Location by A17, AMI_3:27, AMI_3:def 2;

per cases ;

suppose da = d ; :: thesis:

then ( (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = ((Comput (P1,(s1 +* (DataPart s2)),i)) . da) + ((Comput (P1,(s1 +* (DataPart s2)),i)) . db) ) by A12, A37, A41, A64, A65, AMI_3:3;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A19, A67, A71; :: thesis:

end;

suppose da <> d ; :: thesis:

then ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A12, A37, A41, A64, A65, AMI_3:3;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A23, A71; :: thesis:

end;

suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 3 ; :: thesis:

then consider da, db being Data-Location such that
A72: CurInstr (P1,(Comput (P1,s1,i))) = SubFrom (da,db) by AMI_5:10;
A73: IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = SubFrom (da,db) by A72, COMPOS_1:11;
A74: (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . (IC ) = succ (IC (Comput (P1,s1,i))) by A72, AMI_3:4;
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A11, A12, A52, A37, A53, A73, AMI_3:4; :: thesis:
thus IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A11, A12, A32, A52, A37, A53, A73, A74, AMI_3:4; :: thesis:
A75: ( (Comput (P1,(s1 +* (DataPart s2)),i)) . da = (Comput (P2,s2,i)) . da & (Comput (P1,(s1 +* (DataPart s2)),i)) . db = (Comput (P2,s2,i)) . db ) by A27;

now

DataPart p c= p by RELAT_1:59;
then A76: dom (DataPart p) c= dom p by GRFUNC_1:2;
let x be set ; :: thesis:
assume A77: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis:
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then x in Data-Locations by A31, A77;
then reconsider d = x as Data-Location by AMI_3:27, AMI_3:def 2;

per cases ;

suppose A78: da = d ; :: thesis:

then ( (Comput (P1,s1,(i + 1))) . d = ((Comput (P1,s1,i)) . da) - ((Comput (P1,s1,i)) . db) & (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) - ((Comput (P2,s2,i)) . db) ) by A12, A52, A37, A72, A73, AMI_3:4;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A2, A10, A31, A44, A72, A75, A77, A76, A78, A4, AMI_5:19; :: thesis:

end;

suppose da <> d ; :: thesis:

then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A12, A52, A37, A72, A73, AMI_3:4;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A31, A48, A77; :: thesis:

end;

now

let x be set ; :: thesis:
assume A79: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis:
then reconsider d = x as Data-Location by A17, AMI_3:27, AMI_3:def 2;

per cases ;

suppose da = d ; :: thesis:

then ( (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) - ((Comput (P2,s2,i)) . db) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = ((Comput (P1,(s1 +* (DataPart s2)),i)) . da) - ((Comput (P1,(s1 +* (DataPart s2)),i)) . db) ) by A12, A37, A41, A72, A73, AMI_3:4;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A19, A75, A79; :: thesis:

end;

suppose da <> d ; :: thesis:

then ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A12, A37, A41, A72, A73, AMI_3:4;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A23, A79; :: thesis:

end;

suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 4 ; :: thesis:

then consider da, db being Data-Location such that
A80: CurInstr (P1,(Comput (P1,s1,i))) = MultBy (da,db) by AMI_5:11;
A81: IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = MultBy (da,db) by A80, COMPOS_1:11;
A82: (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . (IC ) = succ (IC (Comput (P1,s1,i))) by A80, AMI_3:5;
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A11, A12, A52, A37, A53, A81, AMI_3:5; :: thesis:
thus IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A11, A12, A32, A52, A37, A53, A81, A82, AMI_3:5; :: thesis:
A83: ( (Comput (P1,(s1 +* (DataPart s2)),i)) . da = (Comput (P2,s2,i)) . da & (Comput (P1,(s1 +* (DataPart s2)),i)) . db = (Comput (P2,s2,i)) . db ) by A27;

now

DataPart p c= p by RELAT_1:59;
then A84: dom (DataPart p) c= dom p by GRFUNC_1:2;
let x be set ; :: thesis:
assume A85: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis:
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then x in Data-Locations by A31, A85;
then reconsider d = x as Data-Location by AMI_3:27, AMI_3:def 2;

per cases ;

suppose A86: da = d ; :: thesis:

then ( (Comput (P1,s1,(i + 1))) . d = ((Comput (P1,s1,i)) . da) * ((Comput (P1,s1,i)) . db) & (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db) ) by A12, A52, A37, A80, A81, AMI_3:5;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A2, A10, A31, A44, A80, A83, A85, A84, A86, A4, AMI_5:20; :: thesis:

end;

suppose da <> d ; :: thesis:

then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A12, A52, A37, A80, A81, AMI_3:5;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A31, A48, A85; :: thesis:

end;

now

let x be set ; :: thesis:
assume A87: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis:
then reconsider d = x as Data-Location by A17, AMI_3:27, AMI_3:def 2;

per cases ;

suppose da = d ; :: thesis:

then ( (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) * ((Comput (P2,s2,i)) . db) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = ((Comput (P1,(s1 +* (DataPart s2)),i)) . da) * ((Comput (P1,(s1 +* (DataPart s2)),i)) . db) ) by A12, A37, A41, A80, A81, AMI_3:5;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A19, A83, A87; :: thesis:

end;

suppose da <> d ; :: thesis:

then ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A12, A37, A41, A80, A81, AMI_3:5;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A23, A87; :: thesis:

end;

suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 5 ; :: thesis:

then consider da, db being Data-Location such that
A88: CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) by AMI_5:12;
A89: IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = Divide (da,db) by A88, COMPOS_1:11;
A90: ( (Comput (P1,(s1 +* (DataPart s2)),i)) . da = (Comput (P2,s2,i)) . da & (Comput (P1,(s1 +* (DataPart s2)),i)) . db = (Comput (P2,s2,i)) . db ) by A27;

per cases ;

suppose A91: da <> db ; :: thesis:

A92: (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . (IC ) = succ (IC (Comput (P1,s1,i))) by A88, AMI_3:6;
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A11, A12, A52, A37, A53, A89, AMI_3:6; :: thesis:
thus IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A11, A12, A32, A52, A37, A53, A89, A92, AMI_3:6; :: thesis:

now

DataPart p c= p by RELAT_1:59;
then A93: dom (DataPart p) c= dom p by GRFUNC_1:2;
let x be set ; :: thesis:
assume A94: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis:
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then x in Data-Locations by A31, A94;
then reconsider d = x as Data-Location by AMI_3:27, AMI_3:def 2;

per cases ;

suppose A95: da = d ; :: thesis:

then A96: ( (Comput (P1,s1,(i + 1))) . d = ((Comput (P1,s1,i)) . da) div ((Comput (P1,s1,i)) . db) & (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) div ((Comput (P2,s2,i)) . db) ) by A12, A52, A37, A88, A89, A91, AMI_3:6;
((Comput (P1,(s1 +* (DataPart s2)),i)) . da) div ((Comput (P1,(s1 +* (DataPart s2)),i)) . db) = ((Comput (P1,s1,i)) . da) div ((Comput (P1,s1,i)) . db) by A2, A10, A31, A88, A91, A94, A93, A95, A4, AMI_5:21;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = (Comput (P2,s2,(i + 1))) . d by A90, A94, A96, FUNCT_1:47
.= ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A31, A43, A94, FUNCT_1:47 ;
:: thesis:

end;

suppose A97: db = d ; :: thesis:

then ( (Comput (P1,s1,(i + 1))) . d = ((Comput (P1,s1,i)) . da) mod ((Comput (P1,s1,i)) . db) & (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db) ) by A12, A52, A37, A88, A89, AMI_3:6;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A2, A10, A31, A44, A88, A90, A94, A93, A97, A4, AMI_5:22; :: thesis:

end;

suppose ( da <> d & db <> d ) ; :: thesis:

then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A12, A52, A37, A88, A89, AMI_3:6;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A31, A48, A94; :: thesis:

end;

now

let x be set ; :: thesis:
assume A98: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis:
then reconsider d = x as Data-Location by A17, AMI_3:27, AMI_3:def 2;

per cases ;

suppose da = d ; :: thesis:

then ( (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) div ((Comput (P2,s2,i)) . db) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = ((Comput (P1,(s1 +* (DataPart s2)),i)) . da) div ((Comput (P1,(s1 +* (DataPart s2)),i)) . db) ) by A12, A37, A41, A88, A89, A91, AMI_3:6;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A19, A90, A98; :: thesis:

end;

suppose db = d ; :: thesis:

then ( (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = ((Comput (P1,(s1 +* (DataPart s2)),i)) . da) mod ((Comput (P1,(s1 +* (DataPart s2)),i)) . db) ) by A12, A37, A41, A88, A89, AMI_3:6;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A19, A90, A98; :: thesis:

end;

suppose ( da <> d & db <> d ) ; :: thesis:

then ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A12, A37, A41, A88, A89, AMI_3:6;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A23, A98; :: thesis:

end;

suppose A99: da = db ; :: thesis:

A100: (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . (IC ) = succ (IC (Comput (P1,s1,i))) by A88, AMI_3:6;
hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A11, A12, A52, A37, A53, A89, AMI_3:6; :: thesis:
thus IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A11, A12, A32, A52, A37, A53, A89, A100, AMI_3:6; :: thesis:

now

let x be set ; :: thesis:
assume A101: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis:
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then x in Data-Locations by A31, A101;
then reconsider d = x as Data-Location by AMI_3:27, AMI_3:def 2;

per cases ;

suppose A102: da = d ; :: thesis:

A103: ( ((Comput (P1,s1,i)) | (dom (DataPart p))) . d = (Comput (P1,s1,i)) . d & ((Comput (P2,s2,i)) | (dom (DataPart p))) . d = (Comput (P2,s2,i)) . d ) by A31, A40, A47, A101, FUNCT_1:47;
A104: ( ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . d = (Comput (P1,s1,(i + 1))) . d & ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . d = (Comput (P2,s2,(i + 1))) . d ) by A31, A43, A101, FUNCT_1:47;
(Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db) by A12, A37, A89, A99, A102, AMI_3:6;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A13, A52, A88, A99, A102, A103, A104, AMI_3:6; :: thesis:

end;

suppose da <> d ; :: thesis:

then ( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A12, A52, A37, A88, A89, A99, AMI_3:6;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A31, A48, A101; :: thesis:

end;

now

let x be set ; :: thesis:
assume A105: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis:
then reconsider d = x as Data-Location by A17, AMI_3:27, AMI_3:def 2;

per cases ;

suppose da = d ; :: thesis:

then ( (Comput (P2,s2,(i + 1))) . d = ((Comput (P2,s2,i)) . da) mod ((Comput (P2,s2,i)) . db) & (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = ((Comput (P1,(s1 +* (DataPart s2)),i)) . da) mod ((Comput (P1,(s1 +* (DataPart s2)),i)) . db) ) by A12, A37, A41, A88, A89, A99, AMI_3:6;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A19, A90, A105; :: thesis:

end;

suppose da <> d ; :: thesis:

then ( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A12, A37, A41, A88, A89, A99, AMI_3:6;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A23, A105; :: thesis:

end;

suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 6 ; :: thesis:

then consider loc being Element of NAT such that
A106: CurInstr (P1,(Comput (P1,s1,i))) = SCM-goto loc by AMI_5:13;
A107: CurInstr (P2,(Comput (P2,s2,i))) = SCM-goto (loc + k) by A12, A106, Th1;
thus (IC (Comput (P1,s1,(i + 1)))) + k = loc + k by A52, A106, AMI_3:7
.= IC (Comput (P2,s2,(i + 1))) by A37, A107, AMI_3:7 ; :: thesis:
hence IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A32; :: thesis:

now

let x be set ; :: thesis:
assume A108: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis:
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then x in Data-Locations by A31, A108;
then reconsider d = x as Data-Location by AMI_3:27, AMI_3:def 2;
( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A52, A37, A106, A107, AMI_3:7;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A31, A48, A108; :: thesis:

end;

now

let x be set ; :: thesis:
assume A109: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis:
then reconsider d = x as Data-Location by A17, AMI_3:27, AMI_3:def 2;
( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A37, A41, A106, A107, AMI_3:7;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A23, A109; :: thesis:

end;

suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 7 ; :: thesis:

then consider loc being Element of NAT , da being Data-Location such that
A110: CurInstr (P1,(Comput (P1,s1,i))) = da =0_goto loc by AMI_5:14;

A111: now

per cases ) . da = 0 or (Comput (P1,s1,i)) . da <> 0 ) ;

case (Comput (P1,s1,i)) . da = 0 ; :: thesis:

hence (IC (Comput (P1,s1,(i + 1)))) + k = loc + k by A52, A110, AMI_3:8; :: thesis:

end;

case (Comput (P1,s1,i)) . da <> 0 ; :: thesis:

hence (IC (Comput (P1,s1,(i + 1)))) + k = succ (IC (Comput (P2,s2,i))) by A11, A52, A53, A110, AMI_3:8; :: thesis:

end;

A112: CurInstr (P2,(Comput (P2,s2,i))) = da =0_goto (loc + k) by A12, A110, Th2;

A113: now

per cases ) . da = 0 or (Comput (P2,s2,i)) . da <> 0 ) ;

case (Comput (P2,s2,i)) . da = 0 ; :: thesis:

hence IC (Comput (P2,s2,(i + 1))) = loc + k by A37, A112, AMI_3:8; :: thesis:

end;

case (Comput (P2,s2,i)) . da <> 0 ; :: thesis:

hence IC (Comput (P2,s2,(i + 1))) = succ (IC (Comput (P2,s2,i))) by A37, A112, AMI_3:8; :: thesis:

end;

A114: (Comput (P1,(s1 +* (DataPart s2)),i)) . da = (Comput (P2,s2,i)) . da by A27;

A115: now

per cases ;

suppose loc <> succ (IC (Comput (P1,s1,i))) ; :: thesis:

hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A2, A10, A110, A114, A111, A113, A4, AMI_5:23; :: thesis:

end;

suppose loc = succ (IC (Comput (P1,s1,i))) ; :: thesis:

hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A11, A53, A111, A113; :: thesis:

end;

hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) ; :: thesis:
thus IncAddr ((CurInstr (P1,(Comput (P1,s1,(i + 1))))),k) = CurInstr (P2,(Comput (P2,s2,(i + 1)))) by A32, A115; :: thesis:

now

let x be set ; :: thesis:
assume A116: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis:
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then x in Data-Locations by A31, A116;
then reconsider d = x as Data-Location by AMI_3:27, AMI_3:def 2;
( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A52, A37, A110, A112, AMI_3:8;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A31, A48, A116; :: thesis:

end;

now

let x be set ; :: thesis:
assume A117: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis:
then reconsider d = x as Data-Location by A17, AMI_3:27, AMI_3:def 2;
( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A37, A41, A110, A112, AMI_3:8;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A23, A117; :: thesis:

end;

suppose InsCode (CurInstr (P1,(Comput (P1,s1,i)))) = 8 ; :: thesis:

then consider loc being Element of NAT , da being Data-Location such that
A118: CurInstr (P1,(Comput (P1,s1,i))) = da >0_goto loc by AMI_5:15;

A119: now

per cases ) . da > 0 or (Comput (P1,s1,i)) . da <= 0 ) ;

case (Comput (P1,s1,i)) . da > 0 ; :: thesis:

hence (IC (Comput (P1,s1,(i + 1)))) + k = loc + k by A52, A118, AMI_3:9; :: thesis:

end;

case (Comput (P1,s1,i)) . da <= 0 ; :: thesis:

hence (IC (Comput (P1,s1,(i + 1)))) + k = succ (IC (Comput (P2,s2,i))) by A11, A52, A53, A118, AMI_3:9; :: thesis:

end;

A120: CurInstr (P2,(Comput (P2,s2,i))) = da >0_goto (loc + k) by A12, A118, Th3;

A121: now

per cases ) . da > 0 or (Comput (P2,s2,i)) . da <= 0 ) ;

case (Comput (P2,s2,i)) . da > 0 ; :: thesis:

hence IC (Comput (P2,s2,(i + 1))) = loc + k by A37, A120, AMI_3:9; :: thesis:

end;

case (Comput (P2,s2,i)) . da <= 0 ; :: thesis:

hence IC (Comput (P2,s2,(i + 1))) = succ (IC (Comput (P2,s2,i))) by A37, A120, AMI_3:9; :: thesis:

end;

A122: (Comput (P1,(s1 +* (DataPart s2)),i)) . da = (Comput (P2,s2,i)) . da by A27;

A123: now

per cases ;

suppose loc <> succ (IC (Comput (P1,s1,i))) ; :: thesis:

hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A2, A10, A118, A122, A119, A121, A4, AMI_5:24; :: thesis:

end;

suppose loc = succ (IC (Comput (P1,s1,i))) ; :: thesis:

hence (IC (Comput (P1,s1,(i + 1)))) + k = IC (Comput (P2,s2,(i + 1))) by A11, A53, A119, A121; :: thesis:

end;

now

let x be set ; :: thesis:
assume A124: x in dom ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) ; :: thesis:
dom (DataPart p) c= Data-Locations by RELAT_1:58;
then x in Data-Locations by A31, A124;
then reconsider d = x as Data-Location by AMI_3:27, AMI_3:def 2;
( (Comput (P1,s1,(i + 1))) . d = (Comput (P1,s1,i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A52, A37, A118, A120, AMI_3:9;
hence ((Comput (P1,s1,(i + 1))) | (dom (DataPart p))) . x = ((Comput (P2,s2,(i + 1))) | (dom (DataPart p))) . x by A31, A48, A124; :: thesis:

end;

now

let x be set ; :: thesis:
assume A125: x in dom (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) ; :: thesis:
then reconsider d = x as Data-Location by A17, AMI_3:27, AMI_3:def 2;
( (Comput (P1,(s1 +* (DataPart s2)),(i + 1))) . d = (Comput (P1,(s1 +* (DataPart s2)),i)) . d & (Comput (P2,s2,(i + 1))) . d = (Comput (P2,s2,i)) . d ) by A37, A41, A118, A120, AMI_3:9;
hence (DataPart (Comput (P1,(s1 +* (DataPart s2)),(i + 1)))) . x = (DataPart (Comput (P2,s2,(i + 1)))) . x by A23, A125; :: thesis:

end;

B127: DataPart p c= p by RELAT_1:59;
B130: IC in dom (IncIC (p,k)) by MEMSTR_0:52;

now

thus (IC (Comput (P1,s1,0))) + k = (IC s1) + k by EXTPRO_1:2
.= (IC p) + k by A2, B1, GRFUNC_1:2
.= (IC p) + k
.= IC (IncIC (p,k)) by MEMSTR_0:53
.= IC s2 by A3, B130, GRFUNC_1:2
.= IC (Comput (P2,s2,0)) by EXTPRO_1:2 ; :: thesis:
reconsider loc = IC p as Element of NAT ;
A131: IC p = IC s1 by A2, B1, GRFUNC_1:2;
then IC p = IC (Comput (P1,s1,0)) by EXTPRO_1:2;
then A132: loc in dom q by A8, B2, AMISTD_5:def 4;
A133: (IC p) + k in dom (Reloc (q,k)) by A132, COMPOS_1:46;
B134: IC in dom (IncIC (p,k)) by MEMSTR_0:52;
A135: q . (IC p) = P1 . (IC s1) by A131, A132, A4, GRFUNC_1:2;
dom P2 = NAT by PARTFUN1:def 2;
then A136: CurInstr (P2,(Comput (P2,s2,0))) = P2 . (IC (Comput (P2,s2,0))) by PARTFUN1:def 6
.= P2 . (IC s2) by EXTPRO_1:2
.= P2 . (IC (IncIC (p,k))) by A3, B134, GRFUNC_1:2
.= P2 . ((IC p) + k) by MEMSTR_0:53
.= P2 . ((IC p) + k)
.= (Reloc (q,k)) . ((IC p) + k) by A133, A4, GRFUNC_1:2 ;
A137: dom P1 = NAT by PARTFUN1:def 2;
CurInstr (P1,(Comput (P1,s1,0))) = CurInstr (P1,s1) by EXTPRO_1:2
.= P1 . (IC s1) by A137, PARTFUN1:def 6 ;
hence IncAddr ((CurInstr (P1,(Comput (P1,s1,0)))),k) = CurInstr (P2,(Comput (P2,s2,0))) by A132, A135, A136, COMPOS_1:35; :: thesis:
A138: dom (DataPart s2) = Data-Locations by MEMSTR_0:9;
X1: DataPart p c= s1 by A2, B127, XBOOLE_1:1;
HH: DataPart (IncIC (p,k)) = DataPart p by MEMSTR_0:51;
IncIC (p,k) = IncIC (p,k) ;
then DataPart p c= IncIC (p,k) by HH, MEMSTR_0:12;
then X2: DataPart p c= s2 by A3, XBOOLE_1:1;
thus (Comput (P1,s1,0)) | (dom (DataPart p)) = s1 | (dom (DataPart p)) by EXTPRO_1:2
.= DataPart p by X1, GRFUNC_1:23
.= s2 | (dom (DataPart p)) by X2, GRFUNC_1:23
.= (Comput (P2,s2,0)) | (dom (DataPart p)) by EXTPRO_1:2 ; :: thesis:
thus DataPart (Comput (P1,(s1 +* (DataPart s2)),0)) = DataPart (s1 +* (DataPart s2)) by EXTPRO_1:2
.= DataPart s2 by A138, FUNCT_4:23
.= DataPart (Comput (P2,s2,0)) by EXTPRO_1:2 ; :: thesis:

end;

then A139: S₁[ 0 ] ;
thus for i being Element of NAT holds S₁[i] from NAT_1:sch 1(A139, A126); :: thesis: