set i = a,k1 >=0_goto ((card I) + 1);
set IF = if<0 a,k1,I;
set IsIF = Initialized (stop (if<0 a,k1,I));
if<0 a,k1,I = (Load (a,k1 >=0_goto ((card I) + 1))) ';' I by SCMPDS_4:def 4;
hence if<0 a,k1,I is shiftable ; :: thesis: if<0 a,k1,I is parahalting

now

let s be State of SCMPDS ; :: thesis: ( Initialized (stop (if<0 a,k1,I)) c= s implies ProgramPart b₁ halts_on b₁ )
assume Initialized (stop (if<0 a,k1,I)) c= s ; :: thesis: ProgramPart b₁ halts_on b₁
then A1: s = s +* (Initialized (stop (if<0 a,k1,I))) by FUNCT_4:79;
A2: ( 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 b₁ halts_on b₁

then if<0 a,k1,I is_halting_on s by A2, Th122;
hence ProgramPart s halts_on s by A1, Def3; :: thesis: verum

end;

suppose s . (DataLoc (s . a),k1) >= 0 ; :: thesis: ProgramPart b₁ halts_on b₁

then if<0 a,k1,I is_halting_on s by Th123;
hence ProgramPart s halts_on s by A1, Def3; :: thesis: verum

end;

end;

end;

then Initialized (stop (if<0 a,k1,I)) is halting by AMI_1:def 26;
hence if<0 a,k1,I is parahalting by SCMPDS_4:def 10; :: thesis: verum