let n be Nat; :: thesis: for x, y being Element of REAL n holds (|.(x + y).| ^2) + (|.(x - y).| ^2) = 2 * ((|.x.| ^2) + (|.y.| ^2))
let x, y be Element of REAL n; :: thesis: (|.(x + y).| ^2) + (|.(x - y).| ^2) = 2 * ((|.x.| ^2) + (|.y.| ^2))
( len x = n & len y = n ) by CARD_1:def 7;
hence (|.(x + y).| ^2) + (|.(x - y).| ^2) = 2 * ((|.x.| ^2) + (|.y.| ^2)) by EUCLID_2:13; :: thesis: verum