let n be Nat; :: thesis: for K being Field
for M1 being Matrix of n,K st M1 is Idempotent & M1 is Orthogonal holds
M1 is symmetric
let K be Field; :: thesis: for M1 being Matrix of n,K st M1 is Idempotent & M1 is Orthogonal holds
M1 is symmetric
let M1 be Matrix of n,K; :: thesis: ( M1 is Idempotent & M1 is Orthogonal implies M1 is symmetric )
assume A1: ( M1 is Idempotent & M1 is Orthogonal ) ; :: thesis: M1 is symmetric
then ( M1 is invertible & M1 @ = M1 ~ ) by MATRIX_6:def 7;
then M1 = 1. K,n by A1, Th10;
hence M1 is symmetric ; :: thesis: verum