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

M2 is_less_than M2 + M1

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

A1: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

assume A2: M1 is Positive ; :: thesis: M2 is_less_than M2 + M1

for i, j being Nat st [i,j] in Indices M2 holds

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

M2 is_less_than M2 + M1

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

A1: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

assume A2: M1 is Positive ; :: thesis: M2 is_less_than M2 + M1

for i, j being Nat st [i,j] in Indices M2 holds

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

proof

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

assume A3: [i,j] in Indices M2 ; :: thesis: M2 * (i,j) < (M2 + M1) * (i,j)

then M1 * (i,j) > 0 by A2, A1;

then M2 * (i,j) < (M2 * (i,j)) + (M1 * (i,j)) by XREAL_1:29;

hence M2 * (i,j) < (M2 + M1) * (i,j) by A3, MATRIXR1:25; :: thesis: verum

end;assume A3: [i,j] in Indices M2 ; :: thesis: M2 * (i,j) < (M2 + M1) * (i,j)

then M1 * (i,j) > 0 by A2, A1;

then M2 * (i,j) < (M2 * (i,j)) + (M1 * (i,j)) by XREAL_1:29;

hence M2 * (i,j) < (M2 + M1) * (i,j) by A3, MATRIXR1:25; :: thesis: verum