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=
  width A & len y=len x & len x >0 & len A >0 holds A*(x-y)=A*x - A*y
proof
  let x,y be FinSequence of REAL, A be Matrix of REAL;
  assume that
A1: len x=width A and
A2: len y=len x and
A3: len x >0 and
A4: len A >0;
A5: len ColVec2Mx y=len y by A2,A3,MATRIXR1:def 9;
A6: width ColVec2Mx y=1 by A2,A3,MATRIXR1:def 9;
A7: len ColVec2Mx x=len x by A3,MATRIXR1:def 9;
  then
A8: len (A*(ColVec2Mx x))=len A by A1,MATRIX_3:def 4
    .=len (A*(ColVec2Mx y)) by A1,A2,A5,MATRIX_3:def 4;
A9: width ColVec2Mx x=1 by A3,MATRIXR1:def 9;
  then
A10: 1<=width(A*(ColVec2Mx x)) by A1,A7,MATRIX_3:def 4;
A11: width (A*(ColVec2Mx x))=width (ColVec2Mx (x)) by A1,A7,MATRIX_3:def 4
    .=width (ColVec2Mx y) by A3,A6,MATRIXR1:def 9
    .=width(A*(ColVec2Mx y)) by A1,A2,A5,MATRIX_3:def 4;
  thus A*(x-y)=Col(A*((ColVec2Mx x)-(ColVec2Mx y)),1) by A2,A3,Th24
    .=Col(A*(ColVec2Mx x)-A*(ColVec2Mx y),1) by A1,A2,A3,A4,A7,A5,A9,A6,Th20
    .=A*x - A*y by A8,A11,A10,Th26;
end;
