reserve a,b for Real,
  i,j,n for Nat,
  M,M1,M2,M3,M4 for Matrix of n, REAL;

theorem
  |:M1+M2:| is_less_or_equal_with |:M1:|+|:M2:|
proof
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 st [i,j] in Indices |:M1+M2:| holds |:M1+M2:|*(i,j)<=(|:M1:|+|:
  M2:|)*(i,j)
  proof
    let i,j;
    assume [i,j] in Indices |:M1+M2:|;
    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 thesis by A5,COMPLEX1:56;
  end;
  hence thesis;
end;
