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;

theorem Th13:
  x in dom f implies (g*f).x = g.(f.x)
proof
  assume
A1: x in dom f;
  per cases;
  suppose
    f.x in dom g;
    then x in dom(g*f) by A1,Th11;
    hence thesis by Th12;
  end;
  suppose
A2: not f.x in dom g;
    then not x in dom(g*f) by Th11;
    hence (g*f).x = {} by Def2
      .= g.(f.x) by A2,Def2;
  end;
end;
