reserve X,Y for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,h for Function;

theorem Th2:
  f c= g iff dom f c= dom g &
   for x being object st x in dom f holds f.x = g.x
proof
  thus f c= g implies dom f c= dom g &
   for x being object st x in dom f holds f.x = g.x
  proof
    assume
A1: f c= g;
    hence dom f c= dom g by RELAT_1:11;
    let x be object;
    assume x in dom f;
    then [x,f.x] in f by FUNCT_1:def 2;
    hence thesis by A1,FUNCT_1:1;
  end;
  assume that
A2: dom f c= dom g and
A3: for x being object st x in dom f holds f.x = g.x;
  let x,y be object;
  assume
A4: [x,y] in f;
  then
A5: x in dom f by FUNCT_1:1;
  y = f.x by A4,FUNCT_1:1;
  then y = g.x by A3,A5;
  hence thesis by A2,A5,FUNCT_1:def 2;
end;
