
theorem Th18: :: stolen from WAYBEL13:30
  for L, M being non empty RelStr st L, M are_isomorphic & L is
  complete holds M is complete
proof
  let L, M be non empty RelStr such that
A1: L, M are_isomorphic and
A2: L is complete;
  let X be Subset of M;
  M, L are_isomorphic by A1,WAYBEL_1:6;
  then consider f being Function of M, L such that
A3: f is isomorphic;
  reconsider fX = f.:X as Subset of L;
  consider fa being Element of L such that
A4: fa is_<=_than fX and
A5: for fb being Element of L st fb is_<=_than fX holds fb <= fa by A2;
  set a = (f qua Function)".fa;
A6: rng f = the carrier of L by A3,WAYBEL_0:66;
  then a in dom f by A3,FUNCT_1:32;
  then reconsider a as Element of M;
A7: fa = f.a by A3,A6,FUNCT_1:35;
  take a;
A8: dom f = the carrier of M by FUNCT_2:def 1;
  hereby
    let b be Element of M such that
A9: b in X;
    reconsider fb = f.b as Element of L;
    fb in fX by A8,A9,FUNCT_1:def 6;
    then fa <= fb by A4;
    hence a <= b by A3,A7,WAYBEL_0:66;
  end;
  let b be Element of M such that
A10: b is_<=_than X;
  reconsider fb = f.b as Element of L;
  fb is_<=_than fX
  proof
    let fc be Element of L;
    assume fc in fX;
    then consider c being object such that
A11: c in dom f and
A12: c in X and
A13: fc = f.c by FUNCT_1:def 6;
    reconsider c as Element of M by A11;
    b <= c by A10,A12;
    hence thesis by A3,A13,WAYBEL_0:66;
  end;
  then fb <= fa by A5;
  hence thesis by A3,A7,WAYBEL_0:66;
end;
