reserve x,y for set,
  i,j,k,l,m,n for Nat,
  K for Field,
  N for without_zero finite Subset of NAT,
  a,b for Element of K,
  A,B,B1,B2,X,X1,X2 for (Matrix of K),
  A9 for (Matrix of m,n,K),
  B9 for (Matrix of m,k,K);

theorem Th2:
  1_K * A = A & a * (b * A) = (a * b) * A
proof
  set 1A=1_K*A;
  set bA=b*A;
  set ab=a*b;
  set abA=ab*A;
A1: now
    let i,j such that
A2: [i,j] in Indices A;
    thus A*(i,j) = 1_K*(A*(i,j))
      .= 1A*(i,j) by A2,MATRIX_3:def 5;
  end;
A3: len(a*bA) = len bA by MATRIX_3:def 5
    .= len A by MATRIX_3:def 5
    .= len abA by MATRIX_3:def 5;
A4: now
A5: Indices bA=Indices A by MATRIXR1:18;
A6: Indices (a*bA)=Indices bA by MATRIXR1:18;
    let i,j such that
A7: [i,j] in Indices (a*bA);
    thus (a*bA)*(i,j) = a*(bA*(i,j)) by A7,A6,MATRIX_3:def 5
      .= a*(b*(A*(i,j))) by A7,A6,A5,MATRIX_3:def 5
      .= (a*b)*(A*(i,j)) by GROUP_1:def 3
      .= abA*(i,j) by A7,A6,A5,MATRIX_3:def 5;
  end;
A8: width(a*bA) = width bA by MATRIX_3:def 5
    .= width A by MATRIX_3:def 5
    .= width abA by MATRIX_3:def 5;
  len 1A=len A & width 1A=width A by MATRIX_3:def 5;
  hence thesis by A1,A3,A8,A4,MATRIX_0:21;
end;
