let n be Nat; :: thesis: for M1, M2 being Matrix of n,REAL holds |:(M1 + M2):| is_less_or_equal_with |:M1:| + |:M2:|

let M1, M2 be Matrix of n,REAL; :: thesis: |:(M1 + M2):| is_less_or_equal_with |:M1:| + |:M2:|

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

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

A3: 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) <= (|:M1:| + |:M2:|) * (i,j)

let M1, M2 be Matrix of n,REAL; :: thesis: |:(M1 + M2):| is_less_or_equal_with |:M1:| + |:M2:|

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

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

A3: 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) <= (|:M1:| + |:M2:|) * (i,j)

proof

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

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

then A4: [i,j] in Indices (M1 + M2) by Th5;

then [i,j] in Indices |:M1:| by A1, A2, Th5;

then A5: (|:M1:| + |:M2:|) * (i,j) = (|:M1:| * (i,j)) + (|:M2:| * (i,j)) by MATRIXR1:25

.= |.(M1 * (i,j)).| + (|:M2:| * (i,j)) by A1, A2, A4, Def7

.= |.(M1 * (i,j)).| + |.(M2 * (i,j)).| by A3, A2, A4, Def7 ;

|:(M1 + M2):| * (i,j) = |.((M1 + M2) * (i,j)).| by A4, Def7

.= |.((M1 * (i,j)) + (M2 * (i,j))).| by A1, A2, A4, MATRIXR1:25 ;

hence |:(M1 + M2):| * (i,j) <= (|:M1:| + |:M2:|) * (i,j) by A5, COMPLEX1:56; :: thesis: verum

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

then A4: [i,j] in Indices (M1 + M2) by Th5;

then [i,j] in Indices |:M1:| by A1, A2, Th5;

then A5: (|:M1:| + |:M2:|) * (i,j) = (|:M1:| * (i,j)) + (|:M2:| * (i,j)) by MATRIXR1:25

.= |.(M1 * (i,j)).| + (|:M2:| * (i,j)) by A1, A2, A4, Def7

.= |.(M1 * (i,j)).| + |.(M2 * (i,j)).| by A3, A2, A4, Def7 ;

|:(M1 + M2):| * (i,j) = |.((M1 + M2) * (i,j)).| by A4, Def7

.= |.((M1 * (i,j)) + (M2 * (i,j))).| by A1, A2, A4, MATRIXR1:25 ;

hence |:(M1 + M2):| * (i,j) <= (|:M1:| + |:M2:|) * (i,j) by A5, COMPLEX1:56; :: thesis: verum