let P be Instruction-Sequence of SCMPDS; :: thesis:
let s be State of SCMPDS; :: thesis:
let I be halt-free shiftable Program of ; :: thesis:
let a be Int_position; :: thesis:
let k1 be Integer; :: thesis:
set b = DataLoc ((s . a),k1);
set IF = if>0 (a,k1,I);
set pI = stop I;
set pIF = stop (if>0 (a,k1,I));
set s2 = Initialize s;
set s3 = Initialize s;
set P2 = P +* (stop I);
set P3 = P +* (stop (if>0 (a,k1,I)));
set s4 = Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1);
set P4 = P +* (stop (if>0 (a,k1,I)));
set i = (a,k1) <=0_goto ((card I) + 1);
set SAl = Start-At (((card I) + 1),SCMPDS);
A1: IC (Initialize s) = 0 by MEMSTR_0:47;
A2: not DataLoc ((s . a),k1) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A3: (Initialize s) . (DataLoc (((Initialize s) . a),k1)) = (Initialize s) . (DataLoc ((s . a),k1)) by FUNCT_4:11
.= s . (DataLoc ((s . a),k1)) by A2, FUNCT_4:11 ;
A4: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) by EXTPRO_1:2
.= Exec (((a,k1) <=0_goto ((card I) + 1)),(Initialize s)) by Th3 ;
assume s . (DataLoc ((s . a),k1)) > 0 ; :: thesis:
then A5: IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) = (IC (Initialize s)) + 1 by A4, A3, SCMPDS_2:56
.= 0 + 1 by A1 ;
for a being Int_position holds (Initialize s) . a = (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) . a by A4, SCMPDS_2:56;
then A6: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)) by SCMPDS_4:8;
assume A7: I is_closed_on s,P ; :: thesis:
then A8: I is_closed_on Initialize s,P +* (stop I) ;
A9: ( Start-At (0,SCMPDS) c= Initialize s & Shift ((stop I),1) c= P +* (stop (if>0 (a,k1,I))) ) by Lm6, FUNCT_4:25;
A10: stop I c= P +* (stop I) by FUNCT_4:25;
assume A11: I is_halting_on s,P ; :: thesis:
then A12: P +* (stop I) halts_on Initialize s ;
A13: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
A14: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A13
.= CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) by A8, A9, A5, A6, Th22, A10
.= halt SCMPDS by A12, EXTPRO_1:def 15 ;
then A15: P +* (stop (if>0 (a,k1,I))) halts_on Initialize s by EXTPRO_1:29;
A16: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) = (a,k1) <=0_goto ((card I) + 1) by Th3;

now :: thesis:

let l be Nat; :: thesis:
assume A17: l < (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 ; :: thesis:
A18: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),0) = Initialize s by EXTPRO_1:2;

per cases ;

suppose l = 0 ; :: thesis:

then CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) = CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) by A18;
hence CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) <> halt SCMPDS by A16; :: thesis:

end;

suppose l <> 0 ; :: thesis:

then consider n being Nat such that
A19: l = n + 1 by NAT_1:6;
reconsider n = n as Nat ;
A20: n < LifeSpan ((P +* (stop I)),(Initialize s)) by A17, A19, XREAL_1:6;
assume A21: CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS ; :: thesis:
A22: Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),(n + 1)) = Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),n) by EXTPRO_1:4;
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),n))) = CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),n))) by A8, A9, A5, A6, Th22, A10
.= halt SCMPDS by A19, A21, A22 ;
hence contradiction by A12, A20, EXTPRO_1:def 15; :: thesis:

end;

then for l being Nat st CurInstr ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),l))) = halt SCMPDS holds
(LifeSpan ((P +* (stop I)),(Initialize s))) + 1 <= l ;
then A23: LifeSpan ((P +* (stop (if>0 (a,k1,I)))),(Initialize s)) = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 by A14, A15, EXTPRO_1:def 15;
A24: DataPart (Result ((P +* (stop I)),(Initialize s))) = DataPart (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A12, EXTPRO_1:23
.= DataPart (Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A8, A9, A5, A6, Th22, A10
.= DataPart (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) by EXTPRO_1:4
.= DataPart (Result ((P +* (stop (if>0 (a,k1,I)))),(Initialize s))) by A15, A23, EXTPRO_1:23 ;

A25: now :: thesis:

let x be object ; :: thesis:
A26: dom (Start-At (((card I) + 1),SCMPDS)) = {(IC )} by FUNCOP_1:13;
assume A27: x in dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) ; :: thesis:

per cases by A27, SCMPDS_4:6;

suppose A28: x is Int_position ; :: thesis:

then x <> IC by SCMPDS_2:43;
then A29: not x in dom (Start-At (((card I) + 1),SCMPDS)) by A26, TARSKI:def 1;
thus (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if>0 (a,k1,I)))),(Initialize s))) . x
.= (Result ((P +* (stop I)),(Initialize s))) . x by A24, A28, SCMPDS_4:8
.= (IExec (I,P,(Initialize s))) . x
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . x by A29, FUNCT_4:11 ; :: thesis:

end;

suppose A30: x = IC ; :: thesis:

A31: IC (Result ((P +* (stop I)),(Initialize s))) = IC (IExec (I,P,(Initialize s)))
.= card I by A7, A11, Th25 ;
A32: x in dom (Start-At (((card I) + 1),SCMPDS)) by A26, A30, TARSKI:def 1;
thus (IExec ((if>0 (a,k1,I)),P,(Initialize s))) . x = (Result ((P +* (stop (if>0 (a,k1,I)))),(Initialize s))) . x
.= (Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) . x by A15, A23, EXTPRO_1:23
.= IC (Comput ((P +* (stop (if>0 (a,k1,I)))),(Comput ((P +* (stop (if>0 (a,k1,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A30, EXTPRO_1:4
.= (IC (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s)))))) + 1 by A8, A9, A5, A6, Th22, A10
.= (IC (Result ((P +* (stop I)),(Initialize s)))) + 1 by A12, EXTPRO_1:23
.= IC (Start-At (((card I) + 1),SCMPDS)) by A31, FUNCOP_1:72
.= ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) . x by A30, A32, FUNCT_4:13 ; :: thesis:

end;

dom (IExec ((if>0 (a,k1,I)),P,(Initialize s))) = the carrier of SCMPDS by PARTFUN1:def 2
.= dom ((IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS))) by PARTFUN1:def 2 ;
hence IExec ((if>0 (a,k1,I)),P,(Initialize s)) = (IExec (I,P,(Initialize s))) +* (Start-At (((card I) + 1),SCMPDS)) by A25, FUNCT_1:2; :: thesis: