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 M1 = n & width M1 = n ) by MATRIX_1:25;
A4: ( len M2 = n & width M2 = n ) by MATRIX_1:25;
(M1 + M2) @ = (M1 @) + (M2 @) by Th24
.= (- M1) + (M2 @) by A1, Def6
.= (- M1) + (- M2) by A2, Def6
.= - (M1 + M2) by A3, A4, MATRIX_4:12 ;
hence M1 + M2 is antisymmetric by Def6; :: thesis: verum