let N be Matrix of n,K; :: thesis: ( N = - M implies N is central_symmetric )

assume A1: N = - M ; :: thesis: N is central_symmetric

A2: Indices M = [:(Seg n),(Seg n):] by MATRIX_0:24;

let i, j, k, l be Nat; :: according to MATRIX17:def 3 :: thesis: ( [i,j] in Indices N & k = (n + 1) - i & l = (n + 1) - j implies N * (i,j) = N * (k,l) )

assume that

A3: [i,j] in Indices N and

A4: ( k = (n + 1) - i & l = (n + 1) - j ) ; :: 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, Def3

.= (- M) * (k,l) by A2, A6, MATRIX_3:def 2 ;

hence N * (i,j) = N * (k,l) by A1; :: thesis: verum

assume A1: N = - M ; :: thesis: N is central_symmetric

A2: Indices M = [:(Seg n),(Seg n):] by MATRIX_0:24;

let i, j, k, l be Nat; :: according to MATRIX17:def 3 :: thesis: ( [i,j] in Indices N & k = (n + 1) - i & l = (n + 1) - j implies N * (i,j) = N * (k,l) )

assume that

A3: [i,j] in Indices N and

A4: ( k = (n + 1) - i & l = (n + 1) - j ) ; :: 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, Def3

.= (- M) * (k,l) by A2, A6, MATRIX_3:def 2 ;

hence N * (i,j) = N * (k,l) by A1; :: thesis: verum