reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;

theorem Th27:
  dom f = dom g & (for x st x in dom f holds f.x c= g.x) implies
  product f c= product g
proof
  assume that
A1: dom f = dom g and
A2: for x st x in dom f holds f.x c= g.x;
  let x be object;
  assume x in product f;
  then consider f1 such that
A3: x = f1 and
A4: dom f1 = dom f and
A5: for x being object st x in dom f holds f1.x in f.x by Def5;
  now
    let x be object;
    assume
A6: x in dom g;
    then
A7: f1.x in f.x by A1,A5;
    f.x c= g.x by A1,A2,A6;
    hence f1.x in g.x by A7;
  end;
  hence thesis by A1,A3,A4,Def5;
end;
