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

theorem
  for x,y being FinSequence of REAL, A being Matrix of REAL st 
  len x=len A & len y=len x & len x>0 holds (x-y)*A=x*A - y*A
proof
  let x,y be FinSequence of REAL,A be Matrix of REAL;
  assume that
A1: len x=len A and
A2: len y=len x and
A3: len x>0;
A4: width LineVec2Mx y=len y by MATRIXR1:def 10;
A5: width LineVec2Mx x=len x by MATRIXR1:def 10;
  then
A6: width ((LineVec2Mx x)*A)=width A by A1,MATRIX_3:def 4
    .=width ((LineVec2Mx y)*A) by A1,A2,A4,MATRIX_3:def 4;
A7: len LineVec2Mx y=1 by MATRIXR1:def 10;
A8: len LineVec2Mx x=1 by MATRIXR1:def 10;
  then
A9: 1<=len((LineVec2Mx x)*A) by A1,A5,MATRIX_3:def 4;
A10: len ((LineVec2Mx x)*A)=len LineVec2Mx x by A1,A5,MATRIX_3:def 4
    .=len LineVec2Mx y by A7,MATRIXR1:def 10
    .=len((LineVec2Mx y)*A) by A1,A2,A4,MATRIX_3:def 4;
  thus (x-y)*A=Line(((LineVec2Mx x)-(LineVec2Mx y))*A,1) by A2,Th23
    .=Line((LineVec2Mx x)*A-(LineVec2Mx y)*A,1) by A1,A2,A3,A5,A4,A8,A7,Th16
    .=x*A - y*A by A6,A10,A9,Th25;
end;
