let N be Matrix of n,K; :: thesis: ( N = - M implies N is subsymmetric )
assume A1: N = - M ; :: thesis: N is subsymmetric
A2: Indices M = [:(Seg n),(Seg n):] by MATRIX_0:24;
let i, j, k, l be Nat; :: according to MATRIX17:def 1 :: thesis: ( [i,j] in Indices N & k = (n + 1) - j & l = (n + 1) - i implies N * (i,j) = N * (k,l) )
assume that
A3: [i,j] in Indices N and
A4: ( k = (n + 1) - j & l = (n + 1) - i ) ; :: thesis: N * (i,j) = N * (k,l)
A5: Indices (- M) = [:(Seg n),(Seg n):] by MATRIX_0:24;
then ( i in Seg n & j in Seg n ) by A1, A3, ZFMISC_1:87;
then ( k in Seg n & l in Seg n ) by A4, Lm1;
then A6: [k,l] in [:(Seg n),(Seg n):] by ZFMISC_1:87;
(- M) * (i,j) = - (M * (i,j)) by A1, A2, A3, A5, MATRIX_3:def 2
.= - (M * (k,l)) by A1, A2, A3, A5, A4, Def1
.= (- M) * (k,l) by A2, A6, MATRIX_3:def 2 ;
hence N * (i,j) = N * (k,l) by A1; :: thesis: verum