let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
for I, J being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let s be 0 -started State of SCMPDS; :: thesis: for I, J being halt-free parahalting shiftable Program of SCMPDS
for a being Int_position
for k1 being Integer holds IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let I, J be halt-free parahalting shiftable Program of SCMPDS; :: thesis: for a being Int_position
for k1 being Integer holds IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let a be Int_position ; :: thesis: for k1 being Integer holds IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
let k1 be Integer; :: thesis: IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2
set IF = if=0 (a,k1,I,J);
A1: ( J is_closed_on s,P & J is_halting_on s,P ) by Th34, Th35;
A2: ( I is_closed_on s,P & I is_halting_on s,P ) by Th34, Th35;
I: Initialize s = s by MEMSTR_0:44;

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)),P,s)) = ((card I) + (card J)) + 2

then IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A2, Th54;
hence IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2 by FUNCT_4:113; :: thesis: verum

end;

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

then IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (J,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th55, I;
hence IC (IExec ((if=0 (a,k1,I,J)),P,s)) = ((card I) + (card J)) + 2 by FUNCT_4:113; :: thesis: verum

end;

end;

end;