let s be State of SCMPDS; :: thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s & I is_halting_on s holds
IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))
let I be halt-free shiftable Program of SCMPDS; :: thesis: for a being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s & I is_halting_on s holds
IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))
let a be Int_position ; :: thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s & I is_halting_on s holds
IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))
let k1 be Integer; :: thesis: ( s . (DataLoc ((s . a),k1)) <= 0 & I is_closed_on s & I is_halting_on s implies IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS)) )
set b = DataLoc ((s . a),k1);
set IF = if<=0 (a,k1,I);
set pIF = stop (if<=0 (a,k1,I));
set pI = stop I;
set s2 = (Initialize s) +* (stop 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 2;
set j = goto ((card I) + 1);
set SAl = Start-At (((card I) + 2),SCMPDS);
I1: s +* (Initialize (stop I)) = (Initialize s) +* (stop I) by COMPOS_1:125;
I2: s +* (Initialize (stop (if<=0 (a,k1,I)))) = (Initialize s) +* (stop (if<=0 (a,k1,I))) by COMPOS_1:125;
A1: if<=0 (a,k1,I) = ((a,k1) <=0_goto 2) ';' ((goto ((card I) + 1)) ';' I) by SCMPDS_4:52;
A2: 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 EXTPRO_1:4
.= Following ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I))))) by EXTPRO_1:3
.= Exec (((a,k1) <=0_goto 2),((Initialize s) +* (stop (if<=0 (a,k1,I))))) by A1, Th22, I2 ;
A3: 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 A4: ((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 I2, FUNCT_4:12
.= s . (DataLoc ((s . a),k1)) by A3, I2, FUNCT_4:12 ;
A5: IC ((Initialize s) +* (stop (if<=0 (a,k1,I)))) = 0 by I2, FUNCT_4:26, SCMPDS_5:18;
A6: ( Initialize (stop I) c= (Initialize s) +* (stop I) & Shift ((stop I),2) c= Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1) ) by Lm6, I1, FUNCT_4:26;
A7: DataPart ((Initialize s) +* (stop I)) = DataPart ((Initialize s) +* (stop (if<=0 (a,k1,I)))) by COMPOS_1:138, FUNCT_7:134;

now

let a be Int_position ; :: thesis: ((Initialize s) +* (stop I)) . a = (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) . a
thus ((Initialize s) +* (stop I)) . a = ((Initialize s) +* (stop (if<=0 (a,k1,I)))) . a by A7, SCMPDS_4:23
.= (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) . a by A2, SCMPDS_2:68 ; :: thesis: verum

end;

then A8: DataPart ((Initialize s) +* (stop I)) = DataPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) by SCMPDS_4:23;
assume s . (DataLoc ((s . a),k1)) <= 0 ; :: thesis: ( not I is_closed_on s or not I is_halting_on s or IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS)) )
then A9: 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)))),2) by A2, A4, SCMPDS_2:68
.= 0 + 2 by A5, Th23 ;
A10: dom (ProgramPart s) = NAT by COMPOS_1:34;
assume A11: I is_closed_on s ; :: thesis: ( not I is_halting_on s or IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS)) )
then A12: I is_closed_on (Initialize s) +* (stop I) by Th38;
assume A13: I is_halting_on s ; :: thesis: IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))
then A14: ProgramPart ((Initialize s) +* (stop I)) halts_on (Initialize s) +* (stop I) by Def3;
T: 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;
x: Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),((LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1)) = Comput ((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)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))) by EXTPRO_1:5;
TX4: ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) = ProgramPart (Comput ((ProgramPart (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)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))))) by AMI_1:123;
TX2: ProgramPart ((Initialize s) +* (stop I)) = ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))))) by AMI_1:123;
A15: 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)))),((LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1)))) = CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1))),(Comput ((ProgramPart (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)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))))) by x, T
.= CurInstr ((ProgramPart ((Initialize s) +* (stop I))),(Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))))) by A12, A6, A9, A8, Th45, TX2, TX4
.= halt SCMPDS by A14, EXTPRO_1:def 14 ;
then A16: ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I)))) halts_on (Initialize s) +* (stop (if<=0 (a,k1,I))) by EXTPRO_1:30;
A17: CurInstr ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I))))) = (a,k1) <=0_goto 2 by A1, Th22, I2;

now

let l be Element of NAT ; :: thesis: ( l < (LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 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)))),b₁))) <> halt SCMPDS )
assume A18: l < (LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 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)))),b₁))) <> 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 EXTPRO_1:3;

per cases ( l = 0 or l <> 0 ) ;

suppose l = 0 ; :: 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)))),b₁))) <> halt SCMPDS

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 A17, Th30; :: thesis: verum

end;

suppose l <> 0 ; :: thesis: not 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)))),b₁))) = halt SCMPDS

then consider n being Nat such that
A19: l = n + 1 by NAT_1:6;
reconsider n = n as Element of NAT by ORDINAL1:def 13;
A20: n < LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))) by A18, A19, XREAL_1:8;
assume A21: 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 ; :: thesis: contradiction
T: 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;
x: Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),(n + 1)) = Comput ((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)),n) by EXTPRO_1:5;
TX2: ProgramPart ((Initialize s) +* (stop I)) = ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),n)) by AMI_1:123;
TX4: ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1)) = ProgramPart (Comput ((ProgramPart (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)),n)) by AMI_1:123;
CurInstr ((ProgramPart ((Initialize s) +* (stop I))),(Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),n))) = CurInstr ((ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),1))),(Comput ((ProgramPart (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)),n))) by A12, A6, A9, A8, Th45, TX2, TX4
.= halt SCMPDS by A19, A21, x, T ;
hence contradiction by A14, A20, EXTPRO_1:def 14; :: thesis: verum

end;

end;

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
(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1 <= l ;
then A22: LifeSpan ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I))))) = (LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1 by A15, A16, EXTPRO_1:def 14;
T: 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;
A23: DataPart (Result ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) = DataPart (Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))))) by A14, EXTPRO_1:23
.= DataPart (Comput ((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)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))))) by A12, A6, A9, A8, Th45, T
.= DataPart (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),((LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1))) by EXTPRO_1:5
.= DataPart (Result ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))))) by A16, A22, EXTPRO_1:23 ;

A24: now

let x be set ; :: thesis: ( x in dom (IExec ((if<=0 (a,k1,I)),s)) implies (IExec ((if<=0 (a,k1,I)),s)) . b₁ = ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . b₁ )
A27: dom (Start-At (((card I) + 2),SCMPDS)) = {(IC SCMPDS)} by FUNCOP_1:19;
assume A28: x in dom (IExec ((if<=0 (a,k1,I)),s)) ; :: thesis: (IExec ((if<=0 (a,k1,I)),s)) . b₁ = ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . b₁

per cases ( x is Int_position or x = IC SCMPDS or x is Element of NAT ) by A28, SCMPDS_4:20;

suppose A29: x is Int_position ; :: thesis: (IExec ((if<=0 (a,k1,I)),s)) . b₁ = ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . b₁

then x <> IC SCMPDS by SCMPDS_2:52;
then A30: not x in dom (Start-At (((card I) + 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)),s)) . x = (Result ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))))) . x by FUNCT_4:12
.= (Result ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) . x by A23, A29, SCMPDS_4:23
.= (IExec (I,s)) . x by A31, FUNCT_4:12
.= ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . x by A30, FUNCT_4:12 ;
:: thesis: verum

end;

suppose A32: x = IC SCMPDS ; :: thesis: (IExec ((if<=0 (a,k1,I)),s)) . b₁ = ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . b₁

A33: not x in dom (s | NAT) by A10, A32, COMPOS_1:3;
then A34: IC (Result ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) = IC (IExec (I,s)) by A32, FUNCT_4:12
.= card I by A11, A13, Th48 ;
A35: x in dom (Start-At (((card I) + 2),SCMPDS)) by A27, A32, TARSKI:def 1;
T: 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;
thus (IExec ((if<=0 (a,k1,I)),s)) . x = (Result ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))))) . x by A33, FUNCT_4:12
.= (Comput ((ProgramPart ((Initialize s) +* (stop (if<=0 (a,k1,I))))),((Initialize s) +* (stop (if<=0 (a,k1,I)))),((LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))) + 1))) . x by A16, A22, EXTPRO_1:23
.= IC (Comput ((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)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)))))) by A32, EXTPRO_1:5
.= (IC (Comput ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(LifeSpan ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))))) + 2 by A12, A6, A9, A8, Th45, T
.= (IC (Result ((ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I))))) + 2 by A14, EXTPRO_1:23
.= (Start-At (((card I) + 2),SCMPDS)) . (IC SCMPDS) by A34, FUNCOP_1:87
.= ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . x by A32, A35, FUNCT_4:14 ; :: thesis: verum

end;

suppose x is Element of NAT ; :: thesis: (IExec ((if<=0 (a,k1,I)),s)) . b₁ = ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) . b₁

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

end;

end;

end;

dom (IExec ((if<=0 (a,k1,I)),s)) = the carrier of SCMPDS by PARTFUN1:def 4
.= dom ((IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS))) by PARTFUN1:def 4 ;
hence IExec ((if<=0 (a,k1,I)),s) = (IExec (I,s)) +* (Start-At (((card I) + 2),SCMPDS)) by A24, FUNCT_1:9; :: thesis: verum