 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 n holds
   (Mx2Tran M).:(p+A) = (Mx2Tran M).p + (Mx2Tran M).:A
proof
  set TRn=TOP-REAL n;
  set TRm=TOP-REAL m;
  set MT=Mx2Tran M;
  let A be Subset of TRn;
  A1: dom MT=[#]TRn by FUNCT_2:def 1;
  thus MT.:(p+A)c=MT.p+MT.:A
  proof
   let y be object;
   assume y in MT.:(p+A);
   then consider x be object such that
    x in dom MT and
    A2: x in p+A and
    A3: MT.x=y by FUNCT_1:def 6;
   x in {p+w where w is Element of TRn:w in A} by A2,RUSUB_4:def 8;
   then consider w be Element of TRn such that
    A4: x=p+w & w in A;
   MT.w in MT.:A & MT.x=MT.p+MT.w by A1,A4,Th27,FUNCT_1:def 6;
   then MT.x in {MT.p+u where u is Element of TRm:u in MT.:A};
   hence thesis by A3,RUSUB_4:def 8;
  end;
  let y be object;
  assume y in MT.p+MT.:A;
  then y in {MT.p+u where u is Element of TRm:u in MT.:A} by RUSUB_4:def 8;
  then consider u be Element of TRm such that
   A5: y=MT.p+u and
   A6: u in MT.:A;
  consider w be object such that
   w in dom MT and
   A7: w in A and
   A8: MT.w=u by A6,FUNCT_1:def 6;
  reconsider w as Element of TRn by A7;
  p+w in {p+L where L is Element of TRn:L in A} by A7;
  then A9: p+w in p+A by RUSUB_4:def 8;
  y =MT.(p+w) by A5,A8,Th27;
  hence thesis by A1,A9,FUNCT_1:def 6;
end;
