reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;
reserve X for non empty set,
  Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;

theorem Th49:
  (for y being Element of Y holds g.y = F.(f.y,x)) implies g = F
  [:](f,x)
proof
  assume
A1: for y being Element of Y holds g.y = F.(f.y,x);
  now
    let y be Element of Y;
    thus g.y = F.(f.y,x) by A1
      .= (F[:](f,x)).y by Th48;
  end;
  hence thesis by FUNCT_2:63;
end;
