theorem Th53:
  f|B = g|B implies FinJoin(B,f) = FinJoin(B,g)
proof
  set J = the L_join of L;
A1: Bottom L = the_unity_wrt J by Th18;
  assume
A2: f|B = g|B;
  now
    per cases;
    suppose
A3:   B = {};
      hence FinJoin(B,f) = J$$({}.A,f) .= Bottom L by A1,SETWISEO:31
        .= J$$({}.A,g) by A1,SETWISEO:31
        .= FinJoin(B,g) by A3;
    end;
    suppose
      B <> {};
      hence thesis by A2,Th34;
    end;
  end;
  hence thesis;
end;
