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
  f.:X1 c= f.:X2 & X1 c= dom f & f is one-to-one implies X1 c= X2
proof
  assume that
A1: f.:X1 c= f.:X2 and
A2: X1 c= dom f and
A3: f is one-to-one;
  let x be object;
  assume
A4: x in X1;
  then f.x in f.:X1 by A2,Def6;
  then ex x2 being object st x2 in dom f & x2 in X2 & f.x = f.x2 by A1,Def6;
  hence thesis by A2,A3,A4;
end;
