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

theorem Th46:
  for Q2 being multLoop holds
  for f being homomorphic Function of Q,Q2 holds
  for x,y holds
  y in x * Ker f iff f.x = f.y
proof
  let Q2 be multLoop,f be homomorphic Function of Q,Q2,x,y;
  thus y in x * Ker f implies f.x = f.y
  proof
    assume y in x * Ker f;
    then consider h being Permutation of Q such that
    A1: h in Mlt (Ker f) & y = h.x by Def39;
    f.x = (f*h).x by Th45,A1
    .= f.y by A1,FUNCT_2:15;
    hence thesis;
  end;
  assume A2: f.x = f.y;
  ex h being Permutation of Q st h in Mlt (Ker f) & y = h.x
  proof
    reconsider h = (curry (the multF of Q)).(y / x)
    as Permutation of Q by Th30;
    take h;
    f.(y / x) = f.y / f.x by Th14
    .= 1.Q2 by Th6,A2;
    then A3: y / x in Ker f by Def29;
    h.x = (y / x) * x by FUNCT_5:69
    .= y;
    hence thesis by A3,Th32;
  end;
  hence thesis by Def39;
end;
