reserve a,b,c,k,m,n for Nat;
reserve p for Prime;

theorem Th34:
  not ex m,n being positive Nat st m^2 - n^2 = 1
  proof
    given m,n being positive Nat such that
A1: m^2 - n^2 = 1;
A2: now
      assume m^2 <= n^2;
      then m^2 - n^2 <= n^2 - n^2 by XREAL_1:9;
      then m^2 - n^2 <= 0;
      hence contradiction by A1;
    end;
A3: sqrt(1+n*n) <= sqrt 1 + sqrt (n^2) by SQUARE_1:59;
    sqrt(m^2) = m & sqrt(n^2) = n by SQUARE_1:def 2;
    then m < 1+n or m = 1+n by A1,A3,XXREAL_0:1,SQUARE_1:18;
    then per cases by NAT_1:13;
    suppose m <= n;
      hence thesis by A2,SQUARE_1:15;
    end;
    suppose m = 1+n;
      then (1+n)^2 - n^2 = 1 by A1;
      then 2*n = 0;
      hence thesis;
    end;
  end;
