let IT1, IT2 be odd Element of NAT ; :: thesis: ( ( n is odd & n <= len W & IT1 <= len W & W . IT1 = W . n & ( for k being odd Nat st k <= len W & W . k = W . n holds
IT1 <= k ) & IT2 <= len W & W . IT2 = W . n & ( for k being odd Nat st k <= len W & W . k = W . n holds
IT2 <= k ) implies IT1 = IT2 ) & ( ( not n is odd or not n <= len W ) & IT1 = len W & IT2 = len W implies IT1 = IT2 ) )
hereby :: thesis: ( ( not n is odd or not n <= len W ) & IT1 = len W & IT2 = len W implies IT1 = IT2 )
assume that
n is odd and
n <= len W ; :: thesis: ( IT1 <= len W & W . IT1 = W . n & ( for k being odd Nat st k <= len W & W . k = W . n holds
IT1 <= k ) & IT2 <= len W & W . IT2 = W . n & ( for k being odd Nat st k <= len W & W . k = W . n holds
IT2 <= k ) implies IT1 = IT2 )
assume that
A5: IT1 <= len W and
A6: W . IT1 = W . n and
A7: for k being odd Nat st k <= len W & W . k = W . n holds
IT1 <= k ; :: thesis: ( IT2 <= len W & W . IT2 = W . n & ( for k being odd Nat st k <= len W & W . k = W . n holds
IT2 <= k ) implies IT1 = IT2 )
assume that
A8: IT2 <= len W and
A9: W . IT2 = W . n and
A10: for k being odd Nat st k <= len W & W . k = W . n holds
IT2 <= k ; :: thesis: IT1 = IT2
A11: IT2 <= IT1 by A5, A6, A10;
IT1 <= IT2 by A7, A8, A9;
hence IT1 = IT2 by A11, XXREAL_0:1; :: thesis: verum
end;
thus ( ( not n is odd or not n <= len W ) & IT1 = len W & IT2 = len W implies IT1 = IT2 ) ; :: thesis: verum