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

theorem
  -M1 is_less_or_equal_with M2 implies M1+M2 is Nonnegative
proof
A1: Indices M1=[:Seg n, Seg n:] by MATRIX_0:24;
A2: Indices (-M1)=[:Seg n, Seg n:] by MATRIX_0:24;
A3: Indices (M1+M2)=[:Seg n, Seg n:] by MATRIX_0:24;
  assume
A4: -M1 is_less_or_equal_with M2;
  for i,j st [i,j] in Indices (M1+M2) holds (M1+M2)*(i,j)>=0
  proof
    let i,j;
    assume
A5: [i,j] in Indices (M1+M2);
    then (-M1)*(i,j)<=M2*(i,j) by A4,A2,A3;
    then -M1*(i,j)<=M2*(i,j) by A1,A3,A5,Th2;
    then M1*(i,j)+M2*(i,j)>=0 by XREAL_1:60;
    hence thesis by A1,A3,A5,MATRIXR1:25;
  end;
  hence thesis;
end;
