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

theorem
  dom f = dom g & f c= g implies f = g
proof
  assume that
A1: dom f = dom g and
A2: f c= g;
 for x,y being object holds [x,y] in f iff [x,y] in g
  proof let x,y be object;
    thus [x,y] in f implies [x,y] in g by A2;
    assume
A3: [x,y] in g;
    then x in dom f by A1,XTUPLE_0:def 12;
    then [x,f.x] in f by FUNCT_1:1;
    hence thesis by A2,A3,FUNCT_1:def 1;
  end;
  hence thesis;
end;
