let s be State of SCMPDS; :: thesis:
let I be halt-free shiftable Program of SCMPDS; :: thesis:
let J be shiftable Program of SCMPDS; :: thesis:
let a be Int_position ; :: thesis:
let k1 be Integer; :: thesis:
set b = DataLoc ((s . a),k1);
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; :: thesis:
set i = (a,k1) <>0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set I2 = (I ';' (Goto ((card J) + 1))) ';' J;
set IF = if=0 (a,k1,I,J);
set pI2 = stop ((I ';' (Goto ((card J) + 1))) ';' J);
set s2 = (Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
set s3 = (Initialize s) +* (stop (if=0 (a,k1,I,J)));
set s4 = Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J)))),1);
I1: s +* (Initialize (stop ((I ';' (Goto ((card J) + 1))) ';' J))) = (Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by COMPOS_1:125;
I2: s +* (Initialize (stop (if=0 (a,k1,I,J)))) = (Initialize s) +* (stop (if=0 (a,k1,I,J))) by COMPOS_1:125;
A2: IC ((Initialize s) +* (stop (if=0 (a,k1,I,J)))) = 0 by I2, FUNCT_4:26, SCMPDS_5:18;
A3: not DataLoc ((s . a),k1) in dom (Initialize (stop (if=0 (a,k1,I,J)))) by SCMPDS_4:31;
A4: if=0 (a,k1,I,J) = (((a,k1) <>0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:50
.= ((a,k1) <>0_goto ((card I) + 2)) ';' ((I ';' (Goto ((card J) + 1))) ';' J) by SCMPDS_4:50 ;
then A5: Shift ((stop ((I ';' (Goto ((card J) + 1))) ';' J)),1) c= Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J)))),1) by Lm5;
A6: Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J)))),(0 + 1)) = Following ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),(Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J)))),0))) by EXTPRO_1:4
.= Following ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J))))) by EXTPRO_1:3
.= Exec (((a,k1) <>0_goto ((card I) + 2)),((Initialize s) +* (stop (if=0 (a,k1,I,J))))) by A4, Th22, I2 ;
not a in dom (Initialize (stop (if=0 (a,k1,I,J)))) by SCMPDS_4:31;
then ((Initialize s) +* (stop (if=0 (a,k1,I,J)))) . (DataLoc ((((Initialize s) +* (stop (if=0 (a,k1,I,J)))) . a),k1)) = ((Initialize s) +* (stop (if=0 (a,k1,I,J)))) . (DataLoc ((s . a),k1)) by I2, FUNCT_4:12
.= 0 by A1, A3, I2, FUNCT_4:12 ;
then A7: IC (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J)))),1)) = succ (IC ((Initialize s) +* (stop (if=0 (a,k1,I,J))))) by A6, SCMPDS_2:67
.= 0 + 1 by A2 ;
A8: DataPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))) = DataPart ((Initialize s) +* (stop (if=0 (a,k1,I,J)))) by COMPOS_1:138, FUNCT_7:134;

now

let a be Int_position ; :: thesis:
thus ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))) . a = ((Initialize s) +* (stop (if=0 (a,k1,I,J)))) . a by A8, SCMPDS_4:23
.= (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J)))),1)) . a by A6, SCMPDS_2:67 ; :: thesis:

end;

now

let l be Element of NAT ; :: thesis:
assume A18: l < (LifeSpan ((ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))))) + 1 ; :: thesis:
u: Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J)))),0) = (Initialize s) +* (stop (if=0 (a,k1,I,J))) by EXTPRO_1:3;

per cases ;

suppose l = 0 ; :: thesis:

end;

suppose l <> 0 ; :: thesis:

end;

A24: now

let x be set ; :: thesis:
A27: dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) = {(IC SCMPDS)} by FUNCOP_1:19;
assume A28: x in dom (IExec ((if=0 (a,k1,I,J)),s)) ; :: thesis:

per cases by A28, SCMPDS_4:20;

suppose A29: x is Int_position ; :: thesis:

then x <> IC SCMPDS by SCMPDS_2:52;
then A30: not x in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A27, TARSKI:def 1;
A31: not x in dom (s | NAT) by A10, A29, SCMPDS_2:53;
hence (IExec ((if=0 (a,k1,I,J)),s)) . x = (Result ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J)))))) . x by FUNCT_4:12
.= (Result ((ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))))) . x by A23, A29, SCMPDS_4:23
.= (IExec (((I ';' (Goto ((card J) + 1))) ';' J),s)) . x by A31, FUNCT_4:12
.= ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by A30, FUNCT_4:12 ;
:: thesis:

end;

suppose A32: x = IC SCMPDS ; :: thesis:

end;

suppose x is Element of NAT ; :: thesis:

hence (IExec ((if=0 (a,k1,I,J)),s)) . x = ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) . x by Th26; :: thesis:

end;

dom (IExec ((if=0 (a,k1,I,J)),s)) = the carrier of SCMPDS by PARTFUN1:def 4
.= dom ((IExec (((I ';' (Goto ((card J) + 1))) ';' J),s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS))) by PARTFUN1:def 4 ;
hence IExec ((if=0 (a,k1,I,J)),s) = (IExec (((I ';' (Goto ((card J) + 1))) ';' J),s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A24, FUNCT_1:9
.= ((IExec (I,s)) +* (Start-At ((((card I) + (card J)) + 1),SCMPDS))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A11, A12, Th47
.= (IExec (I,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by Th14 ;
:: thesis: