reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;
reserve e,u for object,
  A for Subset of X;

theorem
  for f,g,h being Function st
  f c= h & g c= h & f misses g holds dom f misses dom g
proof
  let f,g,h be Function such that
A1: f c= h and
A2: g c= h and
A3: f misses g;
  for x being object st x in dom f holds not x in dom g
  proof
    let x be object;
    assume x in dom f;
    then
A4: [x,f.x] in f by Def2;
    now
      assume x in dom g;
      then
A5:   [x,g.x] in g by Def2;
      then f.x = g.x by A1,A2,A4,Def1;
      hence contradiction by A3,A4,A5,XBOOLE_0:3;
    end;
    hence thesis;
  end;
  hence thesis by XBOOLE_0:3;
end;
