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

for M1, M2 being Matrix of n,K st M1 is antisymmetric & M2 is antisymmetric holds

M1 - M2 is antisymmetric

let n be Nat; :: thesis: for M1, M2 being Matrix of n,K st M1 is antisymmetric & M2 is antisymmetric holds

M1 - M2 is antisymmetric

let M1, M2 be Matrix of n,K; :: thesis: ( M1 is antisymmetric & M2 is antisymmetric implies M1 - M2 is antisymmetric )

assume that

A1: M1 is antisymmetric and

A2: M2 is antisymmetric ; :: thesis: M1 - M2 is antisymmetric

A3: ( len (- M2) = n & width (- M2) = n ) by MATRIX_0:24;

A4: ( len M1 = n & width M1 = n ) by MATRIX_0:24;

(M1 - M2) @ = (M1 @) + ((- M2) @) by Th23

.= (- M1) + ((- M2) @) by A1

.= (- M1) + (- (M2 @)) by Th26

.= (- M1) + (- (- M2)) by A2

.= - (M1 - M2) by A3, A4, MATRIX_4:12 ;

hence M1 - M2 is antisymmetric ; :: thesis: verum

for M1, M2 being Matrix of n,K st M1 is antisymmetric & M2 is antisymmetric holds

M1 - M2 is antisymmetric

let n be Nat; :: thesis: for M1, M2 being Matrix of n,K st M1 is antisymmetric & M2 is antisymmetric holds

M1 - M2 is antisymmetric

let M1, M2 be Matrix of n,K; :: thesis: ( M1 is antisymmetric & M2 is antisymmetric implies M1 - M2 is antisymmetric )

assume that

A1: M1 is antisymmetric and

A2: M2 is antisymmetric ; :: thesis: M1 - M2 is antisymmetric

A3: ( len (- M2) = n & width (- M2) = n ) by MATRIX_0:24;

A4: ( len M1 = n & width M1 = n ) by MATRIX_0:24;

(M1 - M2) @ = (M1 @) + ((- M2) @) by Th23

.= (- M1) + ((- M2) @) by A1

.= (- M1) + (- (M2 @)) by Th26

.= (- M1) + (- (- M2)) by A2

.= - (M1 - M2) by A3, A4, MATRIX_4:12 ;

hence M1 - M2 is antisymmetric ; :: thesis: verum