reserve A for set,
  C for non empty set,
  B for Subset of A,
  x for Element of A,
  f,g for Function of A,C;
reserve B for Element of Fin A;
reserve L for non empty LattStr,
  a,b,c for Element of L;
reserve L for Lattice;
reserve a,b,c,u,v for Element of L;
reserve A for non empty set,
  x for Element of A,
  B for Element of Fin A,
  f,g for Function of A, the carrier of L;
reserve L for 0_Lattice,
  f,g for Function of A, the carrier of L,
  u for Element of L;

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;
