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

M1 + M2 is Positive

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

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

A2: ( Indices M1 = [:(Seg n),(Seg n):] & Indices (M1 + M2) = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

assume A3: ( M1 is Nonnegative & M2 is Positive ) ; :: thesis: M1 + M2 is Positive

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

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

M1 + M2 is Positive

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

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

A2: ( Indices M1 = [:(Seg n),(Seg n):] & Indices (M1 + M2) = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

assume A3: ( M1 is Nonnegative & M2 is Positive ) ; :: thesis: M1 + M2 is Positive

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

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

proof

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

assume A4: [i,j] in Indices (M1 + M2) ; :: thesis: (M1 + M2) * (i,j) > 0

then ( M1 * (i,j) >= 0 & M2 * (i,j) > 0 ) by A3, A1, A2;

then (M1 * (i,j)) + (M2 * (i,j)) > 0 ;

hence (M1 + M2) * (i,j) > 0 by A2, A4, MATRIXR1:25; :: thesis: verum

end;assume A4: [i,j] in Indices (M1 + M2) ; :: thesis: (M1 + M2) * (i,j) > 0

then ( M1 * (i,j) >= 0 & M2 * (i,j) > 0 ) by A3, A1, A2;

then (M1 * (i,j)) + (M2 * (i,j)) > 0 ;

hence (M1 + M2) * (i,j) > 0 by A2, A4, MATRIXR1:25; :: thesis: verum