reserve i,j for Nat;

theorem
  for x1,x2 being FinSequence of REAL, A being Matrix of REAL st len x1=
  len x2 & len A=len x1 holds (x1+x2)*A=x1*A + x2*A
proof
  let x1,x2 be FinSequence of REAL,A be Matrix of REAL;
  assume that
A1: len x1=len x2 and
A2: len A=len x1;
A4: width LineVec2Mx x2=len x2 by Def10;
A5: width LineVec2Mx x1=len x1 by Def10;
  then
A6: width ((LineVec2Mx x1)*A)=width A by A2,MATRIX_3:def 4
    .=width ((LineVec2Mx x2)*A) by A1,A2,A4,MATRIX_3:def 4;
A7: len LineVec2Mx x1=1 by Def10;
  then
A8: 1<=len((LineVec2Mx x1)*A) by A2,A5,MATRIX_3:def 4;
A9: len LineVec2Mx x2=1 by Def10;
  thus (x1+x2)*A=Line((LineVec2Mx x1+LineVec2Mx(x2))*A,1) by A1,Th50
    .=Line((LineVec2Mx x1)*A+(LineVec2Mx x2)*A,1) by A1,A2,A5,A4,A7,A9,
MATRIX_4:63
    .=x1*A + x2*A by A6,A8,Th55;
end;
