set i = (a,k1) <>0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set IF = if=0 (a,k1,I,J);
set IsIF = Initialize (stop (if=0 (a,k1,I,J)));
reconsider IJ = (I ';' (Goto ((card J) + 1))) ';' J as shiftable Program of SCMPDS ;
thus if=0 (a,k1,I,J) is shiftable ; :: thesis: if=0 (a,k1,I,J) is parahalting
let s be 0 -started State of SCMPDS; :: according to SCMPDS_4:def 10 :: thesis: ( not stop (if=0 (a,k1,I,J)) c= s or ProgramPart s halts_on s )
AA: Initialize s = s by COMPOS_1:78;
assume stop (if=0 (a,k1,I,J)) c= s ; :: thesis: ProgramPart s halts_on s
then A1: s = (Initialize s) +* (stop (if=0 (a,k1,I,J))) by AA, FUNCT_4:79;
A2: ( J is_closed_on s & J is_halting_on s ) by Th34, Th35;
A3: ( I is_closed_on s & I is_halting_on s ) by Th34, Th35;

per cases ( s . (DataLoc ((s . a),k1)) = 0 or s . (DataLoc ((s . a),k1)) <> 0 ) ;

suppose s . (DataLoc ((s . a),k1)) = 0 ; :: thesis: ProgramPart s halts_on s

then if=0 (a,k1,I,J) is_halting_on s by A3, Th52;
hence ProgramPart s halts_on s by A1, Def3; :: thesis: verum

end;

suppose s . (DataLoc ((s . a),k1)) <> 0 ; :: thesis: ProgramPart s halts_on s

then if=0 (a,k1,I,J) is_halting_on s by A2, Th53;
hence ProgramPart s halts_on s by A1, Def3; :: thesis: verum

end;

end;