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=len
  A & len A=len B & width A=width B & len A >0 holds x*(A-B)=(x*A) - (x*B)
proof
  let x be FinSequence of REAL,A,B be Matrix of REAL;
  assume that
A1: len x=len A and
A2: len A=len B and
A3: width A=width B and
A4: len A >0;
A5: width LineVec2Mx x=len x by MATRIXR1:def 10;
  then
A6: len ((LineVec2Mx x)*A)=len LineVec2Mx x by A1,MATRIX_3:def 4
    .=1 by MATRIXR1:def 10;
A7: len ((LineVec2Mx x)*A)= len LineVec2Mx x by A1,A5,MATRIX_3:def 4
    .= len ((LineVec2Mx x)*B) by A1,A2,A5,MATRIX_3:def 4;
A8: width ((LineVec2Mx x)*A)=width A by A1,A5,MATRIX_3:def 4
    .=width ((LineVec2Mx x)*B) by A1,A2,A3,A5,MATRIX_3:def 4;
  len LineVec2Mx x=1 by MATRIXR1:def 10;
  hence x*(A-B) =Line((LineVec2Mx x)*A-(LineVec2Mx x)*B,1) by A1,A2,A3,A4,A5
,Th20
    .=(x*A) - (x*B) by A8,A7,A6,Th25;
end;
