reserve x,A for set, i,j,k,m,n, l, l1, l2 for Nat;
reserve D for non empty set, z for Nat;
reserve S for COM-Struct;
reserve ins for Element of the InstructionsF of S;
reserve k, m for Nat,
  x, x1, x2, x3, y, y1, y2, y3, X,Y,Z for set;
reserve i, j, k for Nat,
  n for Nat,
  l,il for Nat;
reserve
  i,j,k for Instruction of S,
  I,J,K for Program of S;
reserve k1,k2 for Integer;
reserve l,l1,loc for Nat;
reserve i1,i2 for Instruction of S;
reserve
  i,j,k for Instruction of S,
  I,J,K for Program of S;
reserve m for Nat;

theorem
 for S being COM-Struct
  for P,Q being NAT-defined (the InstructionsF of S)-valued finite Function,
   k being Nat st
  P c= Q holds Reloc(P,k) c= Reloc(Q,k)
proof
 let S be COM-Struct;
  let P,Q be NAT-defined (the InstructionsF of S)-valued finite Function;
  let k be Nat;
  set rP = Reloc(P,k);
  set rQ = Reloc(Q,k);
A1: dom Reloc(P,k) = {m + k where m is Nat: m in dom P} by Th21;
A2: dom Shift(P,k) = {m + k where m is Nat:
       m in dom P} by VALUED_1:def 12;
A3: dom Shift(Q,k) = {m + k where m is Nat:
       m in dom Q} by VALUED_1:def 12;
A4: rQ = IncAddr(Shift(Q,k),k) by Th22;
  assume
A5:  P c= Q;
  then
A6: Shift(P,k) c= Shift(Q,k) by VALUED_1:20;
A7: dom P c= dom Q by A5,GRFUNC_1:2;
A8: now
    let x be object;
    assume x in dom Reloc(P,k);
    then consider m1 being Nat such that
A9: x = m1 + k and
A10: m1 in dom P by A1;
A11:  (m1 + k) in dom Shift(Q,k) by A7,A3,A10;
A12:  (m1 + k) in dom Shift(P,k) by A2,A10;
    then
A13: Shift(P,k)/.(m1+k) = Shift(P,k). (
    m1 + k) by PARTFUN1:def 6
      .= Shift(Q,k). (m1 + k) by A6,A12,GRFUNC_1:2
      .= Shift(Q,k)/.(m1+k) by A11,PARTFUN1:def 6;
    thus (rP).x = IncAddr(Shift(P,k),k).x by Th22
      .= IncAddr(Shift(Q,k)/.(m1+k),k) by A12,A13,A9,Def9
      .= (rQ).x by A9,A11,A4,Def9;
  end;
A14: dom Shift(P,k) c= dom Shift(Q,k) by A6,GRFUNC_1:2;
  now
    let x be object;
    assume x in dom rP;
    then x in dom Shift(P,k) by Th20;
    then x in dom Shift(Q,k) by A14;
    hence x in dom rQ by Th20;
  end;
  then dom rP c= dom rQ;
  hence thesis by A8,GRFUNC_1:2;
end;
