theorem
  for A be Matrix of len b1,len b2,K for B be Matrix of len b2,len B3,K
  st width A = len B for AB be Matrix of len b1,len B3,K st AB = A*B holds
  Mx2Tran(AB,b1,B3) = Mx2Tran(B,b2,B3) * Mx2Tran(A,b1,b2)
proof
  let A be Matrix of len b1,len b2,K;
  let B be Matrix of len b2,len B3,K such that
A1: width A = len B;
  set MB=Mx2Tran(B,b2,B3);
  set MA=Mx2Tran(A,b1,b2);
  let AB be Matrix of len b1,len B3,K such that
A2: AB = A*B;
  set MAB=Mx2Tran(AB,b1,B3);
  now
    let x be object;
    assume x in the carrier of V1;
    then reconsider v=x as Element of V1;
    set L=LineVec2Mx(v|--b1);
A3: len A=len b1 by MATRIX_0:def 2;
A4: width L=len (v|--b1) & len (v|--b1)=len b1 by MATRIX_0:23,MATRLIN:def 7;
    then len L=1 & len (L*A)=len L by A3,MATRIX_0:23,MATRIX_3:def 4;
    then
A5: dom (L*A)=Seg 1 by FINSEQ_1:def 3;
A6: width (L*A)=width A by A4,A3,MATRIX_3:def 4;
    then
A7: len B=len b2 & len Line(L * A,1)=width A by CARD_1:def 7,MATRIX_0:def 2;
A8: 1 in Seg 1;
    dom (MB*MA) = the carrier of V1 by FUNCT_2:def 1;
    hence (MB*MA).x = MB.(MA.v) by FUNCT_1:12
      .= MB.Sum lmlt (Line(L * A,1),b2) by Def3
      .= Sum lmlt (Line(LineVec2Mx(Sum lmlt(Line(L*A,1),b2) |--b2)*B,1),B3)
    by Def3
      .= Sum lmlt (Line(LineVec2Mx(Line(L*A,1))*B,1),B3) by A1,A7,MATRLIN:36
      .= Sum lmlt (Line(LineVec2Mx(Line(L*A*B,1)),1),B3) by A1,A6,A5,A8,Th35
      .= Sum lmlt (Line(LineVec2Mx(Line(L*AB,1)),1),B3) by A1,A2,A4,A3,
MATRIX_3:33
      .= Sum lmlt (Line(L*AB,1),B3) by MATRIX15:25
      .= MAB.x by Def3;
  end;
  hence thesis by FUNCT_2:12;
end;
