 reserve X,Y for set,
         n,m,k,i for Nat,
         r for Real,
         R for Element of F_Real,
         K for Field,
         f,f1,f2,g1,g2 for FinSequence,
         rf,rf1,rf2 for real-valued FinSequence,
         cf,cf1,cf2 for complex-valued FinSequence,
         F for Function;
reserve f,f1,f2 for n-element real-valued FinSequence,
        p,p1,p2 for Point of TOP-REAL n,
        M,M1,M2 for Matrix of n,m,F_Real,
        A,B for Matrix of n,F_Real;

theorem
  for A be Subset of TOP-REAL m holds
   (Mx2Tran M)"(((Mx2Tran M).p)+A) = p + (Mx2Tran M)"A
proof
  set MT=Mx2Tran M;
  set TRn=TOP-REAL n;
  set TRm=TOP-REAL m;
  let A be Subset of TRm;
  set w=p;
  set MTw=MT.w;
  A1: w+MT"A={w+v where v is Element of TRn:v in MT"A} by RUSUB_4:def 8;
  A2: MTw+A={MTw+v where v is Element of TRm:v in A} by RUSUB_4:def 8;
  A3: dom MT=[#]TRn by FUNCT_2:def 1;
  hereby let x be object;
   assume A4: x in MT"(MTw+A);
   then reconsider f=x as Element of TRn;
   MT.x in MTw+A by A4,FUNCT_1:def 7;
   then consider v be Element of TRm such that
    A5: MT.x=MTw+v and
    A6: v in A by A2;
   MT.f-MTw =(v+MTw)-MTw by A5
    .=v+(MTw-MTw) by RLVECT_1:28
    .=v+0.TRm by RLVECT_1:15
    .=v by RLVECT_1:4;
   then v=MT.(f-w) by Th28;
   then A7: f-w in MT"A by A3,A6,FUNCT_1:def 7;
   w+(f-w)=(f-w)+w
    .=f-(w-w) by RLVECT_1:29
    .=f-0.TRn by RLVECT_1:15
    .=f by RLVECT_1:13;
   hence x in w+MT"A by A1,A7;
  end;
  let x be object;
  assume x in w+MT"A;
  then consider v be Element of TRn such that
   A8: x=w+v and
   A9: v in MT"A by A1;
  A10: MT.v in A by A9,FUNCT_1:def 7;
  MT.(w+v)=MTw+MT.v by Th27;
  then MT.x in MTw+A by A2,A8,A10;
  hence thesis by A3,A8,FUNCT_1:def 7;
 end;
