reserve Q,Q1,Q2 for multLoop;
reserve x,y,z,w,u,v for Element of Q;

theorem Th48:
  for Q2 being multLoop holds
  for f being homomorphic Function of Q,Q2 holds
  for x,y holds
  x * lp (Ker f) = y * lp (Ker f) iff f.x = f.y
proof
  let Q2 be multLoop,f be homomorphic Function of Q,Q2;
  A1: for x,y holds f.x = f.y implies x * lp (Ker f) c= y * lp (Ker f)
  proof
    let x,y such that A2: f.x = f.y;
    let z be object;
    assume A3: z in x * lp (Ker f);
    then f.x = f.z by Th47;
    hence z in y * lp (Ker f) by A3,A2,Th47;
  end;
  let x,y;
  x * lp (Ker f) = y * lp (Ker f) implies f.x = f.y
  proof
    assume A4: x * lp (Ker f) = y * lp (Ker f);
    f.y = f.y;
    then y in y * lp (Ker f) by Th47;
    hence thesis by A4,Th47;
  end;
  hence thesis by A1;
end;
