reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th9:
   m<>0 implies 2 * card bool (Seg m\{1}) = card bool (Seg (1+m)\{1})
proof
  assume A1:m<>0;
  set S = Seg (m)\{1},F= bool S;
  {} misses union F;
  then
A2: card UNION(F,{{}}) = card F by Th8;
  {m+1} misses union F
  proof
    assume {m+1} meets union F;
    then consider x such that
A3:   x in {m+1} & x in union F by XBOOLE_0:3;
    x=m+1 & x in S by A3,TARSKI:def 1;
    then m+1 in Seg m by XBOOLE_0:def 5;
    then m+1 <=m by FINSEQ_1:1;
    hence thesis by NAT_1:13;
  end;
  then
A4: card UNION(F,{{m+1}}) = card F by Th8;
  {{}}\/{{m+1}} = {{},{m+1}}=bool{m+1} by ENUMSET1:1,ZFMISC_1:24;
  then
A5: UNION(F,bool{m+1})
     = UNION(F,{{}}) \/ UNION(F,{{m+1}}) by Th7;
  UNION(F,{{}}) misses UNION(F,{{m+1}})
  proof
    assume UNION(F,{{}}) meets UNION(F,{{m+1}});
    then consider x such that
A6:  x in UNION(F,{{}}) & x in UNION(F,{{m+1}}) by XBOOLE_0:3;
    consider a,b be set such that
A7:  a in F & b in {{}} & x=a\/b by A6,SETFAM_1:def 4;
    consider c,d be set such that
A8:  c in F & d in {{m+1}} & x=c\/d by A6,SETFAM_1:def 4;
    b={} & m+1 in {m+1} =d by A7,A8, TARSKI:def 1;
    then m+1 in c\/d =a by A7,A8,XBOOLE_0:def 3;
    then m+1 in Seg m by XBOOLE_0:def 5,A7;
    then m+1 <=m by FINSEQ_1:1;
    hence thesis by NAT_1:13;
  end;
  then
A9: card UNION(F,bool{m+1}) = card UNION(F,{{}}) + card UNION(F,{{m+1}})
  by A5,CARD_2:40;
  1<>1+m by A1;
  then
A10: {1+m}\{1} = {1+m} by ZFMISC_1:14;
  Seg (1+m) = Seg (m) \/ {1+m} by FINSEQ_1:9;
  then Seg (1+m)\{1} = S \/ {1+m} by A10,XBOOLE_1:42;
  hence thesis by A2,A4,A9,Th6;
end;
