
theorem Th11:
  for T1,T2 being TA-structure st the TA-structure of T1 = the
TA-structure of T2 for a1 being adjective of T1, a2 being adjective of T2 st a1
  = a2 holds types a1 = types a2
proof
  let T1,T2 be TA-structure such that
A1: the TA-structure of T1 = the TA-structure of T2;
  let a1 be adjective of T1, a2 be adjective of T2 such that
A2: a1 = a2;
  now
    thus types a1 is Subset of T2 by A1;
    let x be object;
    hereby
      assume x in types a1;
      then consider t1 being type of T1 such that
A3:   x = t1 and
A4:   a1 in adjs t1 by Def12;
      reconsider t2 = t1 as type of T2 by A1;
      adjs t1 = adjs t2 by A1;
      hence ex t2 being type of T2 st x = t2 & a2 in adjs t2 by A2,A3,A4;
    end;
    given t2 being type of T2 such that
A5: x = t2 and
A6: a2 in adjs t2;
    reconsider t1 = t2 as type of T1 by A1;
    adjs t1 = adjs t2 by A1;
    hence x in types a1 by A2,A5,A6,Def12;
  end;
  hence thesis by Def12;
end;
