reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem
  ex x, y st x is irrational & y is irrational & x ^ y is rational
proof
  set w = sqrt 2;
A1: (w ^ w) ^ w = w ^ (w^2) by POWER:33,SQUARE_1:19
    .= w ^ 2 by SQUARE_1:def 2
    .= w^2 by POWER:46
    .= 2 by SQUARE_1:def 2;
  reconsider dwa = 2 as Real;
  per cases;
  suppose
A2: w ^ w is rational;
    take w, w;
    thus thesis by A2,Th1,INT_2:28;
  end;
  suppose
A3: w ^ w is irrational;
    take w ^ w, w;
    thus thesis by A1,A3,Th1,INT_2:28;
  end;
end;
