reserve x for set,
  D for non empty set,
  k,n,m,i,j,l for Nat,
  K for Field;

theorem Th22:
  for A,B,C being Matrix of REAL st len A=len B & width A=width B
& len C = len A & width C = width A & (for i,j being Nat st [i,j] in
  Indices A holds C*(i,j) = A*(i,j) - B*(i,j)) holds C=A-B
proof
  let A,B,C be Matrix of REAL;
  assume that
A1: len A=len B & width A=width B and
A2: len C =len A & width C=width A and
A3: for i,j being Nat st [i,j] in Indices A holds C*(i,j) = A
  *(i,j) - B*(i,j);
A4: Indices B=Indices A by A1,MATRIX_4:55;
  for i,j being Nat st [i,j] in Indices A holds C*(i,j) = A*(i,j) +((-B)*(
  i,j))
  proof
    let i,j be Nat;
    reconsider i0=i,j0=j as Nat;
    assume
A5: [i,j] in Indices A;
    hence C*(i,j) = A*(i0,j0) - B*(i0,j0) by A3
      .=A*(i0,j0) +-B*(i0,j0)
      .=A*(i,j)+(-B)*(i,j) by A4,A5,Th10;
  end;
  then C=(A+-B) by A2,MATRIXR1:26;
  hence thesis by MATRIX_4:def 1;
end;
