reserve D for non empty set,
  i,j,k,l for Nat,
  n for Nat,
  x for set,
  a,b,c,r,r1,r2 for Real,
  p,q for FinSequence of REAL,
  MR,MR1 for Matrix of REAL;

theorem Th43:
  MR is m-nonnegative iff Mx2FinS(MR) is nonnegative
proof
  hereby
    assume
A1: MR is m-nonnegative;
    for k st k in dom(Mx2FinS(MR)) holds (Mx2FinS(MR)).k >= 0
    proof
      let i such that
A2:   i in dom Mx2FinS(MR);
      i in Seg len Mx2FinS(MR) by A2,FINSEQ_1:def 3;
      then i >= 1 by FINSEQ_1:1;
      then reconsider l = i - 1 as Nat by NAT_1:21;
A3:   (Mx2FinS(MR)).i = MR*(((l div width MR)+1),((l mod width MR)+1)) by A2
,Th41;
      [((l div width MR)+1),((l mod width MR)+1)] in Indices MR by A2,Th41;
      hence thesis by A1,A3,MATRPROB:def 6;
    end;
    hence Mx2FinS(MR) is nonnegative;
  end;
  assume
A4: Mx2FinS(MR) is nonnegative;
  now
    let i,j such that
A5: [i,j] in Indices MR;
A6: MR*(i,j) = (Mx2FinS(MR)).((i-1) * (width MR) + j) by A5,Th40;
    (i-1) * (width MR) + j in dom Mx2FinS MR by A5,Th40;
    hence MR*(i,j) >= 0 by A4,A6;
  end;
  hence thesis by MATRPROB:def 6;
end;
