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

theorem Th12:
  for a,b being Real,A being Matrix of REAL holds (a+b)*A=a*A + b* A
proof
  let a,b be Real, A be Matrix of REAL;
A1: len (a*A)=len A & width (a*A)=width A by MATRIXR1:27;
A2: len ((a+b)*A)=len A & width ((a+b)*A)=width A by MATRIXR1:27;
A3: for i,j being Nat st [i,j] in Indices ((a+b)*A) holds ((a+b)*A)*(i,j)=(a
  *A + b*A)*(i,j)
  proof
    let i,j be Nat;
    assume
A4: [i,j] in Indices ((a+b)*A);
    reconsider i0=i,j0=j as Nat;
A5: Indices ((a+b)*A)=Indices A by A2,MATRIX_4:55;
    Indices (a*A)=Indices A by A1,MATRIX_4:55;
    then (a*A + b*A)*(i,j) =(a*A)*(i0,j0)+(b*A)*(i0,j0) by A4,A5,MATRIXR1:25
      .=a*(A*(i0,j0))+(b*A)*(i0,j0) by A4,A5,MATRIXR1:29
      .=a*(A*(i0,j0))+b*(A*(i0,j0)) by A4,A5,MATRIXR1:29
      .=(a+b)*(A*(i,j));
    hence thesis by A4,A5,MATRIXR1:29;
  end;
  len (a*A + b*A) = len (a*A) & width (a*A + b*A) =width (a*A) by
MATRIX_3:def 3;
  hence thesis by A2,A1,A3,MATRIX_0:21;
end;
