let M be Matrix of n,K; :: thesis: ( M = (n,n) --> a implies M is subsymmetric )

assume A1: M = (n,n) --> a ; :: thesis: M is subsymmetric

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

assume that

A2: [i,j] in Indices M and

A3: ( k = (n + 1) - j & l = (n + 1) - i ) ; :: thesis: M * (i,j) = M * (k,l)

A4: Indices M = [:(Seg n),(Seg n):] by MATRIX_0:24;

( k in Seg n & l in Seg n ) by A3, A2, A4, Lm2;

then [k,l] in [:(Seg n),(Seg n):] by ZFMISC_1:87;

then M * (k,l) = a by A1, A4, MATRIX_0:46;

hence M * (i,j) = M * (k,l) by A1, A2, MATRIX_0:46; :: thesis: verum

assume A1: M = (n,n) --> a ; :: thesis: M is subsymmetric

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

assume that

A2: [i,j] in Indices M and

A3: ( k = (n + 1) - j & l = (n + 1) - i ) ; :: thesis: M * (i,j) = M * (k,l)

A4: Indices M = [:(Seg n),(Seg n):] by MATRIX_0:24;

( k in Seg n & l in Seg n ) by A3, A2, A4, Lm2;

then [k,l] in [:(Seg n),(Seg n):] by ZFMISC_1:87;

then M * (k,l) = a by A1, A4, MATRIX_0:46;

hence M * (i,j) = M * (k,l) by A1, A2, MATRIX_0:46; :: thesis: verum