let a, b be Real; :: thesis: for n being Nat

for M1, M2 being Matrix of n,REAL st a <= 0 & b <= a & M1 is Nonpositive & M2 is_less_or_equal_with M1 holds

a * M1 is_less_or_equal_with b * M2

let n be Nat; :: thesis: for M1, M2 being Matrix of n,REAL st a <= 0 & b <= a & M1 is Nonpositive & M2 is_less_or_equal_with M1 holds

a * M1 is_less_or_equal_with b * M2

let M1, M2 be Matrix of n,REAL; :: thesis: ( a <= 0 & b <= a & M1 is Nonpositive & M2 is_less_or_equal_with M1 implies a * M1 is_less_or_equal_with b * M2 )

assume that

A1: ( a <= 0 & b <= a ) and

A2: ( M1 is Nonpositive & M2 is_less_or_equal_with M1 ) ; :: thesis: a * M1 is_less_or_equal_with b * M2

A3: Indices (a * M1) = Indices M1 by MATRIXR1:28;

A4: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

for i, j being Nat st [i,j] in Indices (a * M1) holds

(a * M1) * (i,j) <= (b * M2) * (i,j)

for M1, M2 being Matrix of n,REAL st a <= 0 & b <= a & M1 is Nonpositive & M2 is_less_or_equal_with M1 holds

a * M1 is_less_or_equal_with b * M2

let n be Nat; :: thesis: for M1, M2 being Matrix of n,REAL st a <= 0 & b <= a & M1 is Nonpositive & M2 is_less_or_equal_with M1 holds

a * M1 is_less_or_equal_with b * M2

let M1, M2 be Matrix of n,REAL; :: thesis: ( a <= 0 & b <= a & M1 is Nonpositive & M2 is_less_or_equal_with M1 implies a * M1 is_less_or_equal_with b * M2 )

assume that

A1: ( a <= 0 & b <= a ) and

A2: ( M1 is Nonpositive & M2 is_less_or_equal_with M1 ) ; :: thesis: a * M1 is_less_or_equal_with b * M2

A3: Indices (a * M1) = Indices M1 by MATRIXR1:28;

A4: ( Indices M1 = [:(Seg n),(Seg n):] & Indices M2 = [:(Seg n),(Seg n):] ) by MATRIX_0:24;

for i, j being Nat st [i,j] in Indices (a * M1) holds

(a * M1) * (i,j) <= (b * M2) * (i,j)

proof

hence
a * M1 is_less_or_equal_with b * M2
; :: thesis: verum
let i, j be Nat; :: thesis: ( [i,j] in Indices (a * M1) implies (a * M1) * (i,j) <= (b * M2) * (i,j) )

assume A5: [i,j] in Indices (a * M1) ; :: thesis: (a * M1) * (i,j) <= (b * M2) * (i,j)

then ( M1 * (i,j) <= 0 & M2 * (i,j) <= M1 * (i,j) ) by A2, A4, A3;

then a * (M1 * (i,j)) <= b * (M2 * (i,j)) by A1, XREAL_1:67;

then (a * M1) * (i,j) <= b * (M2 * (i,j)) by A3, A5, Th4;

hence (a * M1) * (i,j) <= (b * M2) * (i,j) by A4, A3, A5, Th4; :: thesis: verum

end;assume A5: [i,j] in Indices (a * M1) ; :: thesis: (a * M1) * (i,j) <= (b * M2) * (i,j)

then ( M1 * (i,j) <= 0 & M2 * (i,j) <= M1 * (i,j) ) by A2, A4, A3;

then a * (M1 * (i,j)) <= b * (M2 * (i,j)) by A1, XREAL_1:67;

then (a * M1) * (i,j) <= b * (M2 * (i,j)) by A3, A5, Th4;

hence (a * M1) * (i,j) <= (b * M2) * (i,j) by A4, A3, A5, Th4; :: thesis: verum