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

for M1, M2 being Matrix of n,REAL st M1 is_less_than M2 & a >= 0 holds

a * M1 is_less_or_equal_with a * M2

let n be Nat; :: thesis: for M1, M2 being Matrix of n,REAL st M1 is_less_than M2 & a >= 0 holds

a * M1 is_less_or_equal_with a * M2

let M1, M2 be Matrix of n,REAL; :: thesis: ( M1 is_less_than M2 & a >= 0 implies a * M1 is_less_or_equal_with a * M2 )

assume that

A1: M1 is_less_than M2 and

A2: a >= 0 ; :: thesis: a * M1 is_less_or_equal_with a * M2

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

A4: 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) <= (a * M2) * (i,j)

for M1, M2 being Matrix of n,REAL st M1 is_less_than M2 & a >= 0 holds

a * M1 is_less_or_equal_with a * M2

let n be Nat; :: thesis: for M1, M2 being Matrix of n,REAL st M1 is_less_than M2 & a >= 0 holds

a * M1 is_less_or_equal_with a * M2

let M1, M2 be Matrix of n,REAL; :: thesis: ( M1 is_less_than M2 & a >= 0 implies a * M1 is_less_or_equal_with a * M2 )

assume that

A1: M1 is_less_than M2 and

A2: a >= 0 ; :: thesis: a * M1 is_less_or_equal_with a * M2

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

A4: 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) <= (a * M2) * (i,j)

proof

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

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

then M1 * (i,j) < M2 * (i,j) by A1, A3;

then a * (M1 * (i,j)) <= a * (M2 * (i,j)) by A2, XREAL_1:64;

then A6: (a * M1) * (i,j) <= a * (M2 * (i,j)) by A3, A5, Th4;

[i,j] in Indices M2 by A4, A5, MATRIX_0:24;

hence (a * M1) * (i,j) <= (a * M2) * (i,j) by A6, Th4; :: thesis: verum

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

then M1 * (i,j) < M2 * (i,j) by A1, A3;

then a * (M1 * (i,j)) <= a * (M2 * (i,j)) by A2, XREAL_1:64;

then A6: (a * M1) * (i,j) <= a * (M2 * (i,j)) by A3, A5, Th4;

[i,j] in Indices M2 by A4, A5, MATRIX_0:24;

hence (a * M1) * (i,j) <= (a * M2) * (i,j) by A6, Th4; :: thesis: verum