reserve N for with_zero set;
reserve N for with_zero set;
reserve x,y,z,A,B for set,
  f,g,h for Function,
  i,j,k for Nat;
reserve S for IC-Ins-separated non empty
     with_non-empty_values AMI-Struct over N,
  s for State of S;
reserve N for non empty with_zero set,
 S for IC-Ins-separated non empty with_non-empty_values AMI-Struct over N,
  s for State of S;
reserve n for Nat;
reserve N for with_zero non empty set;

theorem Th19:
  i <= j implies for S being halting
IC-Ins-separated non empty with_non-empty_values AMI-Struct over N
 for P being (the InstructionsF of S)-valued NAT-defined Function
 for s being State of S st P halts_at IC Comput(P,s,i)
  holds P halts_at IC Comput(P,s,j)
proof
  assume
A1: i <= j;
  let S be halting IC-Ins-separated non empty
     with_non-empty_values AMI-Struct over N,
   p be NAT-defined (the InstructionsF of S)-valued Function;
  let s be State of S;
  assume that
A2: IC Comput(p,s,i) in dom p and
A3: p.IC Comput(p,s,i) = halt S;
A4: CurInstr(p,Comput(p,s,i)) = halt S by A2,A3,PARTFUN1:def 6;
  hence IC Comput(p,s,j) in dom p by A2,A1,Th5;
  thus p.IC Comput(p,s,j) = halt S by A1,A3,A4,Th5;
end;
