reserve i,j for Nat;

theorem
  for x1,x2 being FinSequence of REAL, A being Matrix of REAL st len x1=
  len x2 & width A=len x1 & len x1>0 holds A*(x1+x2)=A*x1 + A*x2
proof
  let x1,x2 be FinSequence of REAL, A be Matrix of REAL;
  assume that
A1: len x1=len x2 and
A2: width A=len x1 and
A3: len x1 >0;
A5: len ColVec2Mx x2=len x2 by A1,A3,Def9;
A6: len ColVec2Mx x1=len x1 by A3,Def9;
  then
A7: len (A*(ColVec2Mx x1))=len A by A2,MATRIX_3:def 4
    .=len (A*(ColVec2Mx x2)) by A1,A2,A5,MATRIX_3:def 4;
A8: width ColVec2Mx x1=1 by A3,Def9;
  then
A9: 1<=width(A*(ColVec2Mx x1)) by A2,A6,MATRIX_3:def 4;
A10: width ColVec2Mx x2=1 by A1,A3,Def9;
  thus A*(x1+x2)=Col(A*(ColVec2Mx x1+ColVec2Mx x2),1) by A1,A3,Th46
    .=Col(A*(ColVec2Mx x1)+A*(ColVec2Mx x2),1) by A1,A2,A6,A5,A8,A10,
MATRIX_4:62
    .=A*x1 + A*x2 by A7,A9,Th54;
end;
