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
  (for X1,X2 holds f.:(X1 \ X2) = f.:X1 \ f.:X2) implies f is one-to-one
proof
  assume
A1: for X1,X2 holds f.:(X1 \ X2) = f.:X1 \ f.:X2;
  given x1,x2 such that
A2: x1 in dom f & x2 in dom f and
A3: f.x1 = f.x2 and
A4: x1 <> x2;
A5: f.:({x1}\{x2}) = f.:{x1} by A4,ZFMISC_1:14;
A6: f.:({x1}\{x2}) = f.:{x1}\f.:{x2} by A1;
  Im(f,x1) = {f.x1} & Im(f,x2) = {f.x2} by A2,Th58;
  hence contradiction by A3,A5,A6,XBOOLE_1:37;
end;
