theorem Th18:
  F is with_the_same_dom & (for n be Nat holds G.n = (F.n)|D)
  implies G is with_the_same_dom
proof
  assume that
A1: F is with_the_same_dom and
A2: for n be Nat holds G.n = (F.n)|D;
  let n,m be Nat;
  G.m = (F.m)|D by A2;
  then
A3: dom(G.m) = dom(F.m) /\ D by RELAT_1:61;
  G.n = (F.n)|D by A2;
  then dom(G.n) = dom(F.n) /\ D by RELAT_1:61;
  hence thesis by A1,A3;
end;
