
theorem Th7:
  for T0,T1 being Tree, t being Element of tree(T0,T1) holds (t =
  {} implies succ t = { t^<*0*>, t^<*1*> }) & (for p being Element of T0 st t =
  <*0*>^p for sp being FinSequence holds sp in succ p iff <*0*>^sp in succ t) &
  for p being Element of T1 st t = <*1*>^p for sp being FinSequence holds sp in
  succ p iff <*1*>^sp in succ t
proof
  let T0,T1 be Tree, t be Element of tree(T0,T1);
  set RT = tree(T0,T1);
  hereby
    assume
A1: t = {};
    {} in T1 & <*1*> = <*1*>^{} by FINSEQ_1:34,TREES_1:22;
    then <*1*> in RT by TREES_3:68;
    then
A2: t^<*1*> in RT by A1,FINSEQ_1:34;
A3: succ t = { t^<*n*> where n is Nat:t^<*n*> in RT } by
TREES_2:def 5;
    {} in T0 & <*0*> = <*0*>^{} by FINSEQ_1:34,TREES_1:22;
    then <*0*> in RT by TREES_3:68;
    then
A4: t^<*0*> in RT by A1,FINSEQ_1:34;
    now
      let x1 be object;
      hereby
        assume x1 in succ t;
        then consider n being Nat such that
A5:     x1 = t^<*n*> and
A6:     t^<*n*> in RT by A3;
        reconsider x = x1 as FinSequence by A5;
        ex p being FinSequence st ( p in T0 & x = <*0*>^p or p in T1 & x =
        <*1*>^p) by A5,A6,TREES_3:68;
        then
A7:     x.1 = 0 or x.1 = 1 by FINSEQ_1:41;
        x1 = <*n*> by A1,A5,FINSEQ_1:34;
        then x = <*0*> or x = <*1*> by A7;
        then x = t^<*0*> or x = t^<*1*> by A1,FINSEQ_1:34;
        hence x1 in { t^<*0*>, t^<*1*> } by TARSKI:def 2;
      end;
      assume x1 in { t^<*0*>, t^<*1*> };
      then x1 = t^<*0*> or x1 = t^<*1*> by TARSKI:def 2;
      hence x1 in succ t by A3,A4,A2;
    end;
    hence succ t = { t^<*0*>, t^<*1*> } by TARSKI:2;
  end;
  hereby
    let p be Element of T0 such that
A8: t = <*0*>^p;
    let sp be FinSequence;
    hereby
      assume sp in succ p;
      then sp in { p^<*n*> where n is Nat : p^<*n*> in T0 } by
TREES_2:def 5;
      then consider n being Nat such that
A9:   sp = p^<*n*> and
A10:  p^<*n*> in T0;
      <*0*>^(p^<*n*>) in RT by A10,TREES_3:69;
      then (<*0*>^p)^<*n*> in RT by FINSEQ_1:32;
      then t^<*n*> in {t^<*k*> where k is Nat : t^<*k*> in RT} by A8;
      then t^<*n*> in succ t by TREES_2:def 5;
      hence <*0*>^sp in succ t by A8,A9,FINSEQ_1:32;
    end;
    set zsp = <*0*>^sp;
    assume zsp in succ t;
    then zsp in {t^<*n*> where n is Nat:t^<*n*> in RT} by
TREES_2:def 5;
    then consider n being Nat such that
A11: zsp = t^<*n*> and
A12: t^<*n*> in RT;
    <*0*>^(p^<*n*>) in RT by A8,A12,FINSEQ_1:32;
    then p^<*n*> in T0 by TREES_3:69;
    then p^<*n*> in { p^<*k*> where k is Nat : p^<*k*> in T0 };
    then
A13: p^<*n*> in succ p by TREES_2:def 5;
    <*0*>^sp = <*0*>^(p^<*n*>) by A8,A11,FINSEQ_1:32;
    hence sp in succ p by A13,FINSEQ_1:33;
  end;
  let p be Element of T1 such that
A14: t = <*1*>^p;
  let sp be FinSequence;
  hereby
    assume sp in succ p;
    then sp in { p^<*n*> where n is Nat : p^<*n*> in T1 } by
TREES_2:def 5;
    then consider n being Nat such that
A15: sp = p^<*n*> and
A16: p^<*n*> in T1;
    <*1*>^(p^<*n*>) in RT by A16,TREES_3:70;
    then (<*1*>^p)^<*n*> in RT by FINSEQ_1:32;
    then
    t^<*n*> in {t^<*k*> where k is Nat : t^<*k*> in RT} by A14;
    then t^<*n*> in succ t by TREES_2:def 5;
    hence <*1*>^sp in succ t by A14,A15,FINSEQ_1:32;
  end;
  set zsp = <*1*>^sp;
  assume zsp in succ t;
  then zsp in {t^<*n*> where n is Nat:t^<*n*> in RT} by
TREES_2:def 5;
  then consider n being Nat such that
A17: zsp = t^<*n*> and
A18: t^<*n*> in RT;
  <*1*>^(p^<*n*>) in RT by A14,A18,FINSEQ_1:32;
  then p^<*n*> in T1 by TREES_3:70;
  then p^<*n*> in { p^<*k*> where k is Nat : p^<*k*> in T1 };
  then
A19: p^<*n*> in succ p by TREES_2:def 5;
  <*1*>^sp = <*1*>^(p^<*n*>) by A14,A17,FINSEQ_1:32;
  hence thesis by A19,FINSEQ_1:33;
end;
