 reserve i,j,k,m,n,m1,n1 for Nat;
 reserve a,r,r1,r2 for Real;
 reserve m0,cn,cd for Integer;
 reserve x1,x2,o for object;
 reserve t for 1_greater Nat;

theorem Lm10:
  card rng(F_dp1(t,a)) in card dom(F_dp1(t,a))
proof
A1: card dom (F_dp1(t,a)) = card (Seg len F_dp1(t,a)) by FINSEQ_1:def 3
   .= card (Seg (t+1)) by Def4 .= t + 1 by FINSEQ_1:57;
      per cases;
        suppose
A3:       rng (F_dp1(t,a)) = Segm t;
          card Segm t in nextcard card Segm t by CARD_1:18; then
          card Segm t in card Segm(t+1) by NAT_1:42;
          hence thesis by A1,A3;
        end;
        suppose
A6:       rng (F_dp1(t,a)) c< Segm t;
          Segm t c= Segm (t+1) by NAT_1:39,XREAL_1:31; then
          rng (F_dp1(t,a)) c< Segm (t+1) by A6,XBOOLE_1:58; then
          card rng (F_dp1(t,a)) in Segm card Segm (t+1) by CARD_2:48;
          hence thesis by A1;
        end;
end;
