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

theorem
  for n,x,y,z being positive Nat holds
  n|^x + n|^y = n|^z iff n = 2 & y = x & z = x+1
  proof
    let n,x,y,z be positive Nat;
    thus n|^x + n|^y = n|^z implies n = 2 & y = x & z = x+1
    proof
      assume
A1:   n|^x + n|^y = n|^z;
      per cases;
      suppose x <= y;
        hence thesis by A1,Lm18;
      end;
      suppose y <= x;
        then n = 2 & y = x & z = y+1 by A1,Lm18;
        hence thesis;
      end;
    end;
    2|^x + 2|^x = 2*2|^x
    .= 2|^(x+1) by NEWTON:6;
    hence thesis;
  end;
