let n be Nat; :: thesis: for M1, M2 being Matrix of n,REAL st M1 is Positive & M2 is Negative & |:M1:| is_less_than |:M2:| holds

M1 + M2 is Negative

let M1, M2 be Matrix of n,REAL; :: thesis: ( M1 is Positive & M2 is Negative & |:M1:| is_less_than |:M2:| implies M1 + M2 is Negative )

assume that

A1: M1 is Positive and

A2: M2 is Negative and

A3: |:M1:| is_less_than |:M2:| ; :: thesis: M1 + M2 is Negative

A4: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;

A5: Indices (M1 + M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;

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

for i, j being Nat st [i,j] in Indices (M1 + M2) holds

(M1 + M2) * (i,j) < 0

M1 + M2 is Negative

let M1, M2 be Matrix of n,REAL; :: thesis: ( M1 is Positive & M2 is Negative & |:M1:| is_less_than |:M2:| implies M1 + M2 is Negative )

assume that

A1: M1 is Positive and

A2: M2 is Negative and

A3: |:M1:| is_less_than |:M2:| ; :: thesis: M1 + M2 is Negative

A4: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;

A5: Indices (M1 + M2) = [:(Seg n),(Seg n):] by MATRIX_0:24;

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

for i, j being Nat st [i,j] in Indices (M1 + M2) holds

(M1 + M2) * (i,j) < 0

proof

hence
M1 + M2 is Negative
; :: thesis: verum
let i, j be Nat; :: thesis: ( [i,j] in Indices (M1 + M2) implies (M1 + M2) * (i,j) < 0 )

assume A7: [i,j] in Indices (M1 + M2) ; :: thesis: (M1 + M2) * (i,j) < 0

then [i,j] in Indices |:M1:| by A4, A5, Th5;

then |:M1:| * (i,j) < |:M2:| * (i,j) by A3;

then |.(M1 * (i,j)).| < |:M2:| * (i,j) by A4, A5, A7, Def7;

then |.(M1 * (i,j)).| < |.(M2 * (i,j)).| by A6, A5, A7, Def7;

then A8: |.(M1 * (i,j)).| - |.(M2 * (i,j)).| < |.(M2 * (i,j)).| - |.(M2 * (i,j)).| by XREAL_1:9;

M2 * (i,j) < 0 by A2, A6, A5, A7;

then A9: - (M2 * (i,j)) = |.(M2 * (i,j)).| by ABSVALUE:def 1;

M1 * (i,j) > 0 by A1, A4, A5, A7;

then |.(M1 * (i,j)).| = M1 * (i,j) by ABSVALUE:def 1;

then (M1 * (i,j)) + (M2 * (i,j)) = |.(M1 * (i,j)).| - |.(M2 * (i,j)).| by A9;

hence (M1 + M2) * (i,j) < 0 by A4, A5, A7, A8, MATRIXR1:25; :: thesis: verum

end;assume A7: [i,j] in Indices (M1 + M2) ; :: thesis: (M1 + M2) * (i,j) < 0

then [i,j] in Indices |:M1:| by A4, A5, Th5;

then |:M1:| * (i,j) < |:M2:| * (i,j) by A3;

then |.(M1 * (i,j)).| < |:M2:| * (i,j) by A4, A5, A7, Def7;

then |.(M1 * (i,j)).| < |.(M2 * (i,j)).| by A6, A5, A7, Def7;

then A8: |.(M1 * (i,j)).| - |.(M2 * (i,j)).| < |.(M2 * (i,j)).| - |.(M2 * (i,j)).| by XREAL_1:9;

M2 * (i,j) < 0 by A2, A6, A5, A7;

then A9: - (M2 * (i,j)) = |.(M2 * (i,j)).| by ABSVALUE:def 1;

M1 * (i,j) > 0 by A1, A4, A5, A7;

then |.(M1 * (i,j)).| = M1 * (i,j) by ABSVALUE:def 1;

then (M1 * (i,j)) + (M2 * (i,j)) = |.(M1 * (i,j)).| - |.(M2 * (i,j)).| by A9;

hence (M1 + M2) * (i,j) < 0 by A4, A5, A7, A8, MATRIXR1:25; :: thesis: verum