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

theorem
  for x being FinSequence of REAL, A,B being Matrix of REAL st 
  len x = width A & len A = len B & width A = width B & 
  len x>0 & len A >0 holds (A-B)*x = A*x - B*x
proof
  let x be FinSequence of REAL,A,B be Matrix of REAL;
  assume that
A1: len x=width A and
A2: len A=len B and
A3: width A=width B and
A4: len x >0 and
A5: len A >0;
A6: len ColVec2Mx x=len x by A4,MATRIXR1:def 9;
  then
A7: len (A*(ColVec2Mx x))=len A by A1,MATRIX_3:def 4
    .=len (B*(ColVec2Mx x)) by A1,A2,A3,A6,MATRIX_3:def 4;
A8: width (A*(ColVec2Mx x))=width (ColVec2Mx x) by A1,A6,MATRIX_3:def 4
    .=1 by A4,MATRIXR1:45;
A9: width (A*(ColVec2Mx x))= width (ColVec2Mx x) by A1,A6,MATRIX_3:def 4
    .= width (B*(ColVec2Mx x)) by A1,A3,A6,MATRIX_3:def 4;
  thus (A-B)*x =Col(A*(ColVec2Mx x)-B*(ColVec2Mx x),1) by A1,A2,A3,A4,A5,A6
,Th16
    .=(A*x) - (B*x) by A7,A9,A8,Th26;
end;
