set i = (a,k1) <=0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
set IF = if>0 (a,k1,I,J);
set pIF = stop (if>0 (a,k1,I,J));
reconsider IJ = (I ';' (Goto ((card J) + 1))) ';' J as shiftable Program of ;
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 7 :: thesis: for b₁ being set holds
( not stop (if>0 (a,k1,I,J)) c= b₁ or b₁ halts_on s )
let P be Instruction-Sequence of SCMPDS; :: thesis: ( not stop (if>0 (a,k1,I,J)) c= P or P halts_on s )
A1: Initialize s = s by MEMSTR_0:44;
assume stop (if>0 (a,k1,I,J)) c= P ; :: thesis: P halts_on s
then A2: P = P +* (stop (if>0 (a,k1,I,J))) by FUNCT_4:98;
A3: ( J is_closed_on s,P & J is_halting_on s,P ) by Th11, Th12;
A4: ( I is_closed_on s,P & I is_halting_on s,P ) by Th11, Th12;

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

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

then if>0 (a,k1,I,J) is_halting_on s,P by A4, Th59;
hence P halts_on s by A1, A2; :: thesis: verum

end;

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

then if>0 (a,k1,I,J) is_halting_on s,P by A3, Th60;
hence P halts_on s by A1, A2; :: thesis: verum

end;

end;