let M be Matrix of n,K; :: thesis: ( M = M1 + M2 implies M is central_symmetric )

assume A1: M = M1 + M2 ; :: thesis: M is central_symmetric

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

assume that

A2: [i,j] in Indices M and

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

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

( (n + 1) - j in Seg n & (n + 1) - i in Seg n ) by A2, A4, Lm2;

then A5: [k,l] in [:(Seg n),(Seg n):] by A3, ZFMISC_1:87;

A6: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

(M1 + M2) * (i,j) = (M1 * (i,j)) + (M2 * (i,j)) by A2, A4, A6, MATRIX_3:def 3

.= (M1 * (k,l)) + (M2 * (i,j)) by A2, A4, A6, A3, Def3

.= (M1 * (k,l)) + (M2 * (k,l)) by A2, A4, A6, A3, Def3

.= (M1 + M2) * (k,l) by A6, A5, MATRIX_3:def 3 ;

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

assume A1: M = M1 + M2 ; :: thesis: M is central_symmetric

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

assume that

A2: [i,j] in Indices M and

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

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

( (n + 1) - j in Seg n & (n + 1) - i in Seg n ) by A2, A4, Lm2;

then A5: [k,l] in [:(Seg n),(Seg n):] by A3, ZFMISC_1:87;

A6: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

(M1 + M2) * (i,j) = (M1 * (i,j)) + (M2 * (i,j)) by A2, A4, A6, MATRIX_3:def 3

.= (M1 * (k,l)) + (M2 * (i,j)) by A2, A4, A6, A3, Def3

.= (M1 * (k,l)) + (M2 * (k,l)) by A2, A4, A6, A3, Def3

.= (M1 + M2) * (k,l) by A6, A5, MATRIX_3:def 3 ;

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