let n be Nat; :: thesis: for M being Matrix of n, REAL holds |:M:| is Nonnegative
let M be Matrix of n, REAL ; :: thesis: |:M:| is Nonnegative
for i, j being Nat st [i,j] in Indices |:M:| holds
|:M:| * i,j >= 0

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices |:M:| implies |:M:| * i,j >= 0 )
assume A1: [i,j] in Indices |:M:| ; :: thesis: |:M:| * i,j >= 0
Indices |:M:| = Indices M by Th5;
then |:M:| * i,j = abs (M * i,j) by A1, Def7;
hence |:M:| * i,j >= 0 by COMPLEX1:132; :: thesis: verum

end;

hence |:M:| is Nonnegative by Def4; :: thesis: verum