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_1:24;
A4: ( len M1 = n & width M1 = n ) by MATRIX_1:24;
(M1 - M2) @ = (M1 @) + ((- M2) @) by Th24
.= (- M1) + ((- M2) @) by A1, Def6
.= (- M1) + (- (M2 @)) by Th27
.= (- M1) + (- (- M2)) by A2, Def6
.= - (M1 - M2) by A3, A4, MATRIX_4:12 ;
hence M1 - M2 is antisymmetric by Def6; :: thesis: verum