reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;
reserve A for (Matrix of D),
  A9 for Matrix of n9,m9,D,
  M9 for Matrix of n9, m9,K,
  nt,nt1,nt2 for Element of n-tuples_on NAT,
  mt,mt1 for Element of m -tuples_on NAT,
  M for Matrix of K;

theorem Th41:
  [:rng nt,rng mt:] c= Indices M implies a * Segm(M,nt,mt) = Segm( a * M,nt,mt)
proof
  set Sa=Segm(a * M,nt,mt);
  set S=Segm(M,nt,mt);
  set aS=a*S;
A1: Indices (a*M)=Indices M by MATRIXR1:18;
A2: Indices Sa=Indices S by MATRIX_0:26;
  assume
A3: [:rng nt,rng mt:] c= Indices M;
  now
    let i,j such that
A4: [i,j] in Indices Sa;
A5: aS*(i,j)=a * (S*(i,j)) by A2,A4,MATRIX_3:def 5;
    reconsider i9=i,j9=j as Element of NAT by ORDINAL1:def 12;
A6: Sa*(i9,j9)=(a*M)*(nt.i,mt.j) by A4,Def1;
A7: S*(i9,j9)=M*(nt.i,mt.j) by A2,A4,Def1;
    [nt.i9,mt.j9] in Indices M by A3,A1,A4,Th17;
    hence Sa*(i,j)=aS*(i,j) by A6,A5,A7,MATRIX_3:def 5;
  end;
  hence thesis by MATRIX_0:27;
end;
