reserve i,j for Nat;

theorem Th64:
  for x being FinSequence of REAL, A,B being Matrix of REAL st len
  A=len B & width A=width B & len A=len x 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 A=len B & width A=width B and
A2: len A=len x;
A4: width LineVec2Mx x=len x by Def10;
  then
A5: len ((LineVec2Mx x)*A)=len LineVec2Mx x by A2,MATRIX_3:def 4
    .=1 by Def10;
A6: width ((LineVec2Mx x)*A)=width A by A2,A4,MATRIX_3:def 4
    .=width ((LineVec2Mx x)*B) by A1,A2,A4,MATRIX_3:def 4;
  thus x*(A+B) =Line((LineVec2Mx x)*A+(LineVec2Mx x)*B,1) by A1,A2,A4,
MATRIX_4:62
    .=x*A +x*B by A6,A5,Th55;
end;
