let n be Nat; :: thesis: for K being Field

for M1 being Matrix of n,K st M1 is Self_Reversible & M1 is symmetric holds

M1 is Orthogonal

let K be Field; :: thesis: for M1 being Matrix of n,K st M1 is Self_Reversible & M1 is symmetric holds

M1 is Orthogonal

let M1 be Matrix of n,K; :: thesis: ( M1 is Self_Reversible & M1 is symmetric implies M1 is Orthogonal )

assume A1: ( M1 is Self_Reversible & M1 is symmetric ) ; :: thesis: M1 is Orthogonal

then A2: M1 = M1 @ by MATRIX_6:def 5;

( M1 is invertible & M1 = M1 ~ ) by A1;

hence M1 is Orthogonal by A2, MATRIX_6:def 7; :: thesis: verum

for M1 being Matrix of n,K st M1 is Self_Reversible & M1 is symmetric holds

M1 is Orthogonal

let K be Field; :: thesis: for M1 being Matrix of n,K st M1 is Self_Reversible & M1 is symmetric holds

M1 is Orthogonal

let M1 be Matrix of n,K; :: thesis: ( M1 is Self_Reversible & M1 is symmetric implies M1 is Orthogonal )

assume A1: ( M1 is Self_Reversible & M1 is symmetric ) ; :: thesis: M1 is Orthogonal

then A2: M1 = M1 @ by MATRIX_6:def 5;

( M1 is invertible & M1 = M1 ~ ) by A1;

hence M1 is Orthogonal by A2, MATRIX_6:def 7; :: thesis: verum