
theorem Th40:
  for T being non empty non void reflexive transitive TA-structure
for t being type of T for v1,v2 being FinSequence of the adjectives of T st v1^
v2 is_applicable_to t holds v1 is_applicable_to t & v2 is_applicable_to v1 ast
  t
proof
  let T be non empty non void reflexive transitive TA-structure;
  let t be type of T;
  let v1,v2 be FinSequence of the adjectives of T;
  set v = v1^v2;
A1: apply(v,t) = apply(v1,t)$^apply(v2, v1 ast t) by Th34;
A2: len apply(v2, v1 ast t) = len v2+1 by Def19;
  assume
A3: for i being Nat, a being adjective of T, s being type of
  T st i in dom v & a = v.i & s = apply(v,t).i holds a is_applicable_to s;
  hereby
A4: dom v1 c= dom v by FINSEQ_1:26;
    let i be Nat, a be adjective of T, s be type of T such that
A5: i in dom v1 and
A6: a = v1.i and
A7: s = apply(v1,t).i;
A8: a = v.i by A5,A6,FINSEQ_1:def 7;
    s = apply(v,t).i by A5,A7,Th35;
    hence a is_applicable_to s by A3,A5,A4,A8;
  end;
  let i be Nat, a be adjective of T, s be type of T such that
A9: i in dom v2 and
A10: a = v2.i and
A11: s = apply(v2, v1 ast t).i;
A12: a = v.(len v1+i) by A9,A10,FINSEQ_1:def 7;
  i >= 1 by A9,FINSEQ_3:25;
  then consider j being Nat such that
A13: i = 1+j by NAT_1:10;
  i <= len v2 by A9,FINSEQ_3:25;
  then j < len v2 by A13,NAT_1:13;
  then
A14: j < len apply(v2, v1 ast t) by A2,NAT_1:13;
  len apply(v1,t) = len v1+1 by Def19;
  then len v1+i = len apply(v1,t)+j by A13;
  then s = apply(v,t).(len v1+i) by A11,A1,A13,A14,Th33;
  hence thesis by A3,A9,A12,FINSEQ_1:28;
end;
