reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;

theorem Th7:
  A is finite implies Day A is finite
proof
  assume A1: A is finite;
  defpred P[Nat] means Day $1 is finite;
  A2:P[0] by SURREAL0:2;
  A3:for n be Nat holds P[n] implies P[n+1]
  proof
let n be Nat;
    assume A4:P[n];set n1=n+1;
    Day n1 c= [:bool Day n,bool Day n:]
    proof
      let o1,o2 be object;
      assume A5: [o1,o2] in Day n1;
      reconsider o1,o2 as set by TARSKI:1;
      A6: o1 c= o1 \/ o2 & o2 c= o1 \/ o2 by XBOOLE_1:7;
      o1 \/ o2 c= Day n
      proof
        let a be object;
        assume a in o1\/o2;
        then consider O such that
        A7: O in n1 & a in Day O by A5,SURREAL0:46;
        A8: O in Segm n1 by A7;
        then reconsider O as Nat;
        O < n1 by A8,NAT_1:44;
        then O <=n by NAT_1:13;
        then Segm O c= Segm n by NAT_1:39;
        then Day O c= Day n by SURREAL0:35;
        hence thesis by A7;
      end;
      then o1 c= Day n & o2 c= Day n by A6,XBOOLE_1:1;
      hence thesis by ZFMISC_1:87;
    end;
    hence thesis by A4;
  end;
  for n be Nat holds P[n] from NAT_1:sch 2(A2,A3);
  hence thesis by A1;
end;
