
theorem Th61:
  for T being non empty non void reflexive transitive
TAS-structure for t being type of T, v1,v2 being FinSequence of the adjectives
of T st v1 is_properly_applicable_to t & v2 is_properly_applicable_to v1 ast t
  holds v1^v2 is_properly_applicable_to t
proof
  let T be non empty non void reflexive transitive TAS-structure;
  let t be type of T;
  let v1,v2 be FinSequence of the adjectives of T;
  set v = v1^v2;
  assume
A1: for i being Nat, a being adjective of T, s being type of
  T st i in dom v1 & a = v1.i & s = apply(v1,t).i holds a
  is_properly_applicable_to s;
  assume
A2: for i being Nat, a being adjective of T, s being type of
  T st i in dom v2 & a = v2.i & s = apply(v2, v1 ast t).i holds a
  is_properly_applicable_to s;
A3: apply(v,t) = apply(v1,t)$^apply(v2, v1 ast t) by Th34;
  let i be Nat, a be adjective of T, s be type of T such that
A4: i in dom v and
A5: a = v.i and
A6: s = apply(v, t).i;
A7: i >= 1 by A4,FINSEQ_3:25;
A8: i <= len v by A4,FINSEQ_3:25;
  per cases;
  suppose
    i <= len v1;
    then
A9: i in dom v1 by A7,FINSEQ_3:25;
    then
A10: a = v1.i by A5,FINSEQ_1:def 7;
    s = apply(v1,t).i by A6,A9,Th35;
    hence thesis by A1,A9,A10;
  end;
  suppose
    i > len v1;
    then i >= 1+len v1 by NAT_1:13;
    then consider j being Nat such that
A11: i = len v1+1+j by NAT_1:10;
A12: len apply(v2, v1 ast t) = len v2+1 by Def19;
A13: len v = len v1+len v2 by FINSEQ_1:22;
A14: len apply(v1,t) = len v1+1 by Def19;
    i = len v1+(j+1) by A11;
    then
A15: j+1 <= len v2 by A8,A13,XREAL_1:6;
    then j < len v2 by NAT_1:13;
    then j < len apply(v2, v1 ast t) by A12,NAT_1:13;
    then
A16: s = apply(v2,v1 ast t).(1+j) by A6,A3,A11,A14,Th33;
    j+1 >= 1 by NAT_1:11;
    then
A17: j+1 in dom v2 by A15,FINSEQ_3:25;
    len v1+(j+1) = len apply(v1,t)+j by A14;
    then a = v2.(1+j) by A5,A11,A17,FINSEQ_1:def 7;
    hence thesis by A2,A17,A16;
  end;
end;
