let s be State of SCMPDS ; :: thesis: for I being Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc (s . a),k1) <> 0 holds
IExec (if=0 a,k1,I),s = s +* (Start-At ((card I) + 1),SCMPDS )
let I be Program of SCMPDS ; :: thesis: for a being Int_position
for k1 being Integer st s . (DataLoc (s . a),k1) <> 0 holds
IExec (if=0 a,k1,I),s = s +* (Start-At ((card I) + 1),SCMPDS )
let a be Int_position ; :: thesis: for k1 being Integer st s . (DataLoc (s . a),k1) <> 0 holds
IExec (if=0 a,k1,I),s = s +* (Start-At ((card I) + 1),SCMPDS )
let k1 be Integer; :: thesis: ( s . (DataLoc (s . a),k1) <> 0 implies IExec (if=0 a,k1,I),s = s +* (Start-At ((card I) + 1),SCMPDS ) )
set b = DataLoc (s . a),k1;
set IF = if=0 a,k1,I;
set pIF = stop (if=0 a,k1,I);
set s3 = (Initialize s) +* (stop (if=0 a,k1,I));
set s4 = Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),1;
set i = a,k1 <>0_goto ((card I) + 1);
set SAl = Start-At ((card I) + 1),SCMPDS ;
I1: (Initialize s) +* (stop (if=0 a,k1,I)) = s +* (Initialize (stop (if=0 a,k1,I))) by SCMPDS_4:5;
A1: IC ((Initialize s) +* (stop (if=0 a,k1,I))) = 0 by I1, FUNCT_4:26, SCMPDS_5:18;
A2: not DataLoc (s . a),k1 in dom (Initialize (stop (if=0 a,k1,I))) by SCMPDS_4:31;
not a in dom (Initialize (stop (if=0 a,k1,I))) by SCMPDS_4:31;
then A3: ((Initialize s) +* (stop (if=0 a,k1,I))) . (DataLoc (((Initialize s) +* (stop (if=0 a,k1,I))) . a),k1) = ((Initialize s) +* (stop (if=0 a,k1,I))) . (DataLoc (s . a),k1) by I1, FUNCT_4:12
.= s . (DataLoc (s . a),k1) by A2, I1, FUNCT_4:12 ;
A4: Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),(0 + 1) = Following (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),0 ) by AMI_1:14
.= Following (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))) by AMI_1:13
.= Exec (a,k1 <>0_goto ((card I) + 1)),((Initialize s) +* (stop (if=0 a,k1,I))) by Th22, I1 ;
assume s . (DataLoc (s . a),k1) <> 0 ; :: thesis: IExec (if=0 a,k1,I),s = s +* (Start-At ((card I) + 1),SCMPDS )
then A5: IC (Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),1) = ICplusConst ((Initialize s) +* (stop (if=0 a,k1,I))),((card I) + 1) by A4, A3, SCMPDS_2:67
.= 0 + ((card I) + 1) by A1, Th23 ;
Initialize (stop (if=0 a,k1,I)) c= (Initialize s) +* (stop (if=0 a,k1,I)) by I1, FUNCT_4:26;
then A6: stop (if=0 a,k1,I) c= Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),1 by AMI_1:81, SCMPDS_4:57;
Y: (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),1)) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),1)) = (Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),1) . (IC (Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),1)) by COMPOS_1:38;
TX: ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I))) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),1) by AMI_1:123;
A7: card (if=0 a,k1,I) = (card I) + 1 by Th15;
then (card I) + 1 in dom (stop (if=0 a,k1,I)) by Th25;
then (Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),1) . ((card I) + 1) = (stop (if=0 a,k1,I)) . ((card I) + 1) by A6, GRFUNC_1:8
.= halt SCMPDS by A7, Th25 ;
then A8: CurInstr (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),1) = halt SCMPDS by A5, Y, TX;
then A9: ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I))) halts_on (Initialize s) +* (stop (if=0 a,k1,I)) by AMI_1:146;
A10: CurInstr (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))) = a,k1 <>0_goto ((card I) + 1) by Th22, I1;

now

let l be Element of NAT ; :: thesis: ( l < 1 implies CurInstr (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),l) <> halt SCMPDS )
u: Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),0 = (Initialize s) +* (stop (if=0 a,k1,I)) by AMI_1:13;
assume l < 1 ; :: thesis: CurInstr (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),l) <> halt SCMPDS
then l < 1 + 0 ;
then l = 0 by NAT_1:13;
then CurInstr (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),l) = CurInstr (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))) by u;
hence CurInstr (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),l) <> halt SCMPDS by A10, Th29; :: thesis: verum

end;

then for l being Element of NAT st CurInstr (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),(Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),l) = halt SCMPDS holds
1 <= l ;
then LifeSpan (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))) = 1 by A8, A9, AMI_1:def 46;
then A11: Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),1 = Result (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))) by A9, AMI_1:122;
A12: dom (ProgramPart s) = NAT by COMPOS_1:34;

A13: now

A14: dom (Start-At ((card I) + 1),SCMPDS ) = {(IC SCMPDS )} by FUNCOP_1:19;
let x be set ; :: thesis: ( x in dom (IExec (if=0 a,k1,I),s) implies (IExec (if=0 a,k1,I),s) . b₁ = (s +* (Start-At ((card I) + 1),SCMPDS )) . b₁ )
assume A16: x in dom (IExec (if=0 a,k1,I),s) ; :: thesis: (IExec (if=0 a,k1,I),s) . b₁ = (s +* (Start-At ((card I) + 1),SCMPDS )) . b₁

per cases ( x is Int_position or x = IC SCMPDS or x is Element of NAT ) by A16, SCMPDS_4:20;

suppose A17: x is Int_position ; :: thesis: (IExec (if=0 a,k1,I),s) . b₁ = (s +* (Start-At ((card I) + 1),SCMPDS )) . b₁

then x <> IC SCMPDS by SCMPDS_2:52;
then A18: not x in dom (Start-At ((card I) + 1),SCMPDS ) by A14, TARSKI:def 1;
not x in dom (s | NAT ) by A12, A17, SCMPDS_2:53;
hence (IExec (if=0 a,k1,I),s) . x = (Comput (ProgramPart ((Initialize s) +* (stop (if=0 a,k1,I)))),((Initialize s) +* (stop (if=0 a,k1,I))),1) . x by A11, FUNCT_4:12
.= ((Initialize s) +* (stop (if=0 a,k1,I))) . x by A4, A17, SCMPDS_2:67
.= s . x by A17, SCMPDS_5:19
.= (s +* (Start-At ((card I) + 1),SCMPDS )) . x by A18, FUNCT_4:12 ;
:: thesis: verum

end;

suppose A19: x = IC SCMPDS ; :: thesis: (IExec (if=0 a,k1,I),s) . b₁ = (s +* (Start-At ((card I) + 1),SCMPDS )) . b₁

not x in dom (s | NAT ) by A12, A19, COMPOS_1:3;
hence (IExec (if=0 a,k1,I),s) . x = (card I) + 1 by A5, A11, A19, FUNCT_4:12
.= (s +* (Start-At ((card I) + 1),SCMPDS )) . x by A19, FUNCT_4:121 ;
:: thesis: verum

end;

suppose x is Element of NAT ; :: thesis: (IExec (if=0 a,k1,I),s) . b₁ = (s +* (Start-At ((card I) + 1),SCMPDS )) . b₁

hence (IExec (if=0 a,k1,I),s) . x = (s +* (Start-At ((card I) + 1),SCMPDS )) . x by Th27; :: thesis: verum

end;

end;

end;

dom (IExec (if=0 a,k1,I),s) = the carrier of SCMPDS by PARTFUN1:def 4
.= dom (s +* (Start-At ((card I) + 1),SCMPDS )) by PARTFUN1:def 4 ;
hence IExec (if=0 a,k1,I),s = s +* (Start-At ((card I) + 1),SCMPDS ) by A13, FUNCT_1:9; :: thesis: verum