let s be State of SCMPDS ; :: thesis: for I, J being parahalting shiftable No-StopCode Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec (if=0 a,k1,I,J),s) = inspos (((card I) + (card J)) + 2)
let I, J be parahalting shiftable No-StopCode Program of SCMPDS ; :: thesis: for a being Int_position
for k1 being Integer holds IC (IExec (if=0 a,k1,I,J),s) = inspos (((card I) + (card J)) + 2)
let a be Int_position ; :: thesis: for k1 being Integer holds IC (IExec (if=0 a,k1,I,J),s) = inspos (((card I) + (card J)) + 2)
let k1 be Integer; :: thesis: IC (IExec (if=0 a,k1,I,J),s) = inspos (((card I) + (card J)) + 2)
set IF = if=0 a,k1,I,J;
A1: ( I is_closed_on s & I is_halting_on s ) by Th34, Th35;
A2: ( J is_closed_on s & J is_halting_on s ) by Th34, Th35;

hereby :: thesis: verum

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

suppose s . (DataLoc (s . a),k1) = 0 ; :: thesis: IC (IExec (if=0 a,k1,I,J),s) = inspos (((card I) + (card J)) + 2)

then IExec (if=0 a,k1,I,J),s = (IExec I,s) +* (Start-At (inspos (((card I) + (card J)) + 2))) by A1, Th54;
hence IC (IExec (if=0 a,k1,I,J),s) = inspos (((card I) + (card J)) + 2) by AMI_1:111; :: thesis: verum

end;

suppose s . (DataLoc (s . a),k1) <> 0 ; :: thesis: IC (IExec (if=0 a,k1,I,J),s) = inspos (((card I) + (card J)) + 2)

then IExec (if=0 a,k1,I,J),s = (IExec J,s) +* (Start-At (inspos (((card I) + (card J)) + 2))) by A2, Th55;
hence IC (IExec (if=0 a,k1,I,J),s) = inspos (((card I) + (card J)) + 2) by AMI_1:111; :: thesis: verum

end;

end;

end;