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

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