let s be State of SCMPDS; :: thesis:
let I, 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 pIF = stop (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);
A2: s +* (Initialize (stop ((I ';' (Goto ((card J) + 1))) ';' J))) = (Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by COMPOS_1:125;
A3: s +* (Initialize (stop (if=0 (a,k1,I,J)))) = (Initialize s) +* (stop (if=0 (a,k1,I,J))) by COMPOS_1:125;
A4: IC ((Initialize s) +* (stop (if=0 (a,k1,I,J)))) = 0 by A3, FUNCT_4:26, SCMPDS_5:18;
A5: not DataLoc ((s . a),k1) in dom (Initialize (stop (if=0 (a,k1,I,J)))) by SCMPDS_4:31;
A6: 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 A7: 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 Lm6;
A8: 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 A6, Th22, A3 ;
A9: 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 A9, 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 A8, SCMPDS_2:67 ; :: thesis:

end;

now

let k be Element of NAT ; :: thesis:

per cases ;

suppose 0 < k ; :: thesis:

then consider k1 being Nat such that
A19: k1 + 1 = k by NAT_1:6;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def 13;
reconsider m = IC (Comput ((ProgramPart ((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)))),((Initialize s) +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),k1)) as Element of NAT ;
A20: card (stop (if=0 (a,k1,I,J))) = (card (stop ((I ';' (Goto ((card J) + 1))) ';' J))) + 1 by A18, SCMPDS_5:7;
m in dom (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by A16, Def2;
then m < card (stop ((I ';' (Goto ((card J) + 1))) ';' J)) by AFINSQ_1:70;
then A21: m + 1 < card (stop (if=0 (a,k1,I,J))) by A20, XREAL_1:8;
A22: ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J)))) = ProgramPart (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J)))),1)) by AMI_1:123;
IC (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J)))),k)) = IC (Comput ((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)))),1)),k1)) by A19, EXTPRO_1:5
.= m + 1 by A17, A7, A11, A10, Th45, A22 ;
hence IC (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J)))),k)) in dom (stop (if=0 (a,k1,I,J))) by A21, AFINSQ_1:70; :: thesis:

end;

suppose k = 0 ; :: thesis:

hence IC (Comput ((ProgramPart ((Initialize s) +* (stop (if=0 (a,k1,I,J))))),((Initialize s) +* (stop (if=0 (a,k1,I,J)))),k)) in dom (stop (if=0 (a,k1,I,J))) by A12, A4, EXTPRO_1:3; :: thesis:

end;