reserve m,j,p,q,n,l for Element of NAT;

theorem :: SCMFSA8C:11
  for P,Q being Function, k being Nat st P c= Q holds Shift(P
  ,k) c= Shift(Q,k)
proof
  let P,Q be Function;
  let k be Nat;
  assume
A1: P c= Q;
  then
A2: dom P c= dom Q by GRFUNC_1:2;
A3: dom Shift(P,k) = {m + k where m is Nat: m in dom P} by Def12;
A4: dom Shift(Q,k) = {m + k where m is Nat: m in dom Q} by Def12;
A5: now
    let x be object;
    assume x in dom Shift(P,k);
    then ex m1 being Nat st x = m1 + k & m1 in dom P by A3;
    hence x in dom Shift(Q,k) by A2,A4;
  end;
  now
    let x be object;
    assume x in dom Shift(P,k);
    then consider m1 being Nat such that
A6: x = m1 + k and
A7: m1 in dom P by A3;
    thus Shift(P,k).x = Shift(P,k).(m1 + k) by A6
      .= P.m1 by A7,Def12
      .= Q.m1 by A1,A7,GRFUNC_1:2
      .= Shift(Q,k).(m1 + k) by A2,A7,Def12
      .= Shift(Q,k).x by A6;
  end;
  hence thesis by A5,GRFUNC_1:2,TARSKI:def 3;
end;
