reserve i,j for Nat;

theorem Th52:
  for x being FinSequence of REAL, A being Matrix of REAL st len A
  >0 & width A>0 & (len A=len x or width (A@)=len x) holds (A@)*x = x*A
proof
  let x be FinSequence of REAL,A be Matrix of REAL;
  assume that
A1: len A>0 and
A2: width A>0 and
A3: len A=len x or width (A@)=len x;
A4: len A=len x by A2,A3,MATRIX_0:54;
A5: len A=width (A@) by A2,MATRIX_0:54;
  then len ColVec2Mx x=len x by A1,A3,Def9;
  then
A6: width ((A@)*(ColVec2Mx x))=width ColVec2Mx x by A3,A5,MATRIX_3:def 4;
  width ColVec2Mx x=1 by A1,A3,A5,Def9;
  then
A7: 1 in Seg width ((A@)*(ColVec2Mx x)) by A6,FINSEQ_1:1;
A8: len LineVec2Mx x=1 by Def10;
A9: width LineVec2Mx x=len x by Def10;
  then width LineVec2Mx x=len A by A2,A3,MATRIX_0:54;
  then len ((LineVec2Mx x)*A)=len LineVec2Mx x & width ((LineVec2Mx x)*A)=
  width A by MATRIX_3:def 4;
  then Line((LineVec2Mx x)*A,1) = Line((((LineVec2Mx x)*A)@)@,1) by A2,A8,
MATRIX_0:57
    .= Line(((A@)*((LineVec2Mx x)@))@,1) by A2,A4,A9,MATRIX_3:22
    .= Line(((A@)*(ColVec2Mx x))@,1) by A1,A4,Th49
    .= Col((A@)*(ColVec2Mx x),1) by A7,MATRIX_0:59;
  hence thesis;
end;
