let K be Field; :: thesis: for n being Nat
for M1, M2 being Matrix of n,K st n > 0 & 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 n > 0 & M1 is antisymmetric & M2 is antisymmetric holds
M1 - M2 is antisymmetric
let M1, M2 be Matrix of n,K; :: thesis: ( n > 0 & M1 is antisymmetric & M2 is antisymmetric implies M1 - M2 is antisymmetric )
assume A1: ( n > 0 & M1 is antisymmetric & M2 is antisymmetric ) ; :: thesis: M1 - M2 is antisymmetric
A2: ( len (- M2) = n & width (- M2) = n ) by MATRIX_1:25;
A3: ( len M1 = n & width M1 = n ) by MATRIX_1:25;
(M1 - M2) @ = (M1 @ ) + ((- M2) @ ) by Th24
.= (- M1) + ((- M2) @ ) by A1, Def6
.= (- M1) + (- (M2 @ )) by Th27
.= (- M1) + (- (- M2)) by A1, Def6
.= - (M1 - M2) by A2, A3, MATRIX_4:12 ;
hence M1 - M2 is antisymmetric by Def6; :: thesis: verum