 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 Th24:
  (Mx2Tran M).(f1-f2) = (Mx2Tran M).f1 - (Mx2Tran M).f2
proof
  f1-f2=f1+-f2 by RVSUM_1:31
   .=f1+(-1)*f2 by RVSUM_1:54;
  hence (Mx2Tran M).(f1-f2)=(Mx2Tran M).f1+(Mx2Tran M).((-1)*f2) by Th22
   .=(Mx2Tran M).f1+(-1)*(Mx2Tran M).f2 by Th23
   .=(Mx2Tran M).f1+-(Mx2Tran M).f2 by RVSUM_1:54
   .=(Mx2Tran M).f1-(Mx2Tran M).f2 by RVSUM_1:31;
end;
