reserve k, n for Nat;

theorem Th2:
  for S being non void non empty ManySortedSign, X being
non-empty ManySortedSet of the carrier of S, v being SortSymbol of S, n st 1<=n
holds (ex t being Element of (the Sorts of FreeMSA X).v st depth t = n) iff (ex
c being directed Chain of InducedGraph S st len c = n & (vertex-seq c).(len c +
  1) = v)
proof
  let S be non void non empty ManySortedSign, X be non-empty ManySortedSet
  of the carrier of S, v be SortSymbol of S, n;
  assume
A1: 1<=n;
  set G = InducedGraph S;
  FreeMSA X = MSAlgebra(#FreeSort(X),FreeOper(X)#)by MSAFREE:def 14;
  then
A2: (the Sorts of FreeMSA X).v = FreeSort(X,v) by MSAFREE:def 11;
A3: FreeSort(X,v) c= S-Terms X by MSATERM:12;
  thus (ex t being Element of (the Sorts of FreeMSA X).v st depth t = n)
implies ex c being directed Chain of InducedGraph S st len c = n & (vertex-seq
  c).(len c +1) = v
  proof
    reconsider n1=n as Element of NAT by ORDINAL1:def 12;
    set D = (the carrier' of G)*;
    given t being Element of (the Sorts of FreeMSA X).v such that
A4: depth t = n;
    t in FreeSort(X,v) by A2;
    then reconsider t9 = t as Term of S, X by A3;
    consider dt being finite DecoratedTree, tr being finite Tree such that
A5: dt = t & tr = dom dt and
A6: depth t = height tr by MSAFREE2:def 14;
    not t is root by A1,A4,A5,A6,TREES_1:42,TREES_9:def 1;
    then consider o being OperSymbol of S such that
A7: t9.{} =[o, the carrier of S] by MSSCYC_1:20;
    consider a being ArgumentSeq of Sym(o,X) such that
A8: t = [o,the carrier of S]-tree a by A7,MSATERM:10;
    set args = the_arity_of o;
A9: dom a = dom args by MSATERM:22;
    consider p being FinSequence of NAT such that
A10: p in tr and
A11: len p = height tr by TREES_1:def 12;
    consider i being Nat, T being DecoratedTree, q being Node of T
    such that
A12: i < len a and
    T = a.(i+1) and
A13: p = <*i*>^q by A1,A4,A5,A6,A10,A11,A8,CARD_1:27,TREES_4:11;
    defpred P[Nat, set, set] means ex t1, t2 being Term of S, X st t1 = t|(p|
$1) & t2 = t|(p|($1+1)) & ex o being OperSymbol of S, rs1 being SortSymbol of S
, Ck being Element of D st Ck = $2 & $3 = <*[o,rs1]*>^Ck & [o,the carrier of S]
    = t1.{} & rs1 = the_sort_of t2 & [o,rs1] in the carrier' of G;
    1<=i+1 & i+1<=len a by A12,NAT_1:11,13;
    then
A14: i+1 in dom args by A9,FINSEQ_3:25;
    then reconsider rs1 = args.(i+1) as SortSymbol of S by DTCONSTR:2;
    set e1 = [o,rs1];
A15: (the Arity of S).o = the_arity_of o by MSUALG_1:def 1;
    then
A16: [o,rs1] in InducedEdges S by A14,Def1;
    then reconsider E9 = the carrier' of G as non empty set;
    reconsider e19 = e1 as Element of E9 by A14,A15,Def1;
    reconsider C19 = <*e19*> as Element of D by FINSEQ_1:def 11;
A17: for k being Nat st 1 <= k & k < n1 for x being Element of
    D ex y being Element of D st P[k,x,y]
    proof
      let k be Nat;
      set pk9 = p/^k;
      k<=k+1 by NAT_1:13;
      then
A18:  Seg k c= Seg (k+1) by FINSEQ_1:5;
      k in NAT by ORDINAL1:def 12;
      then
      reconsider pk = p|k, pk1 = p|(k+1) as Node of t by A5,A10,MSSCYC_1:19;
      assume that
      1 <= k and
A19:  k < n1;
A20:  len pk9 = n - k by A4,A6,A11,A19,RFINSEQ:def 1;
      then
A21:  len pk9 <> 0 by A19;
      then
A22:  1 in dom pk9 by CARD_1:27,FINSEQ_5:6;
      reconsider t1 = t9|pk, t2 = t9|pk1 as Term of S, X by MSATERM:29;
      p = pk^pk9 by RFINSEQ:8;
      then
A23:  pk9 in tr|pk by A5,A10,TREES_1:def 6;
      then
A24:  pk9 in dom t1 by A5,TREES_2:def 10;
A25:  k+1<=n by A19,NAT_1:13;
      then
A26:  1<=n-k by XREAL_1:19;
      now
        assume t1 is root;
        then dom t1 = elementary_tree 0 by TREES_9:def 1;
        hence contradiction by A20,A26,A24,TREES_1:42,def 12;
      end;
      then consider o being OperSymbol of S such that
A27:  t1.{} =[o, the carrier of S] by MSSCYC_1:20;
      consider a being ArgumentSeq of Sym(o,X) such that
A28:  t1 = [o,the carrier of S]-tree a by A27,MSATERM:10;
A29:  pk9|1 = <*pk9.1*> by A21,CARD_1:27,FINSEQ_5:20;
      consider i being Nat, T being DecoratedTree, q being Node of
      T such that
A30:  i < len a and
      T = a.(i+1) and
A31:  pk9 = <*i*>^q by A21,A24,A28,CARD_1:27,TREES_4:11;
      reconsider pk9 as Node of t1 by A5,A23,TREES_2:def 10;
      reconsider p1 = pk9|(0+1) as Node of t1 by MSSCYC_1:19;
      reconsider t29 = t1|p1 as Term of S, X;
A32:  p|(k+1)|k = p|(k+1)|Seg k by FINSEQ_1:def 16
        .= p|Seg (k+1)|Seg k by FINSEQ_1:def 16
        .= p|Seg k by A18,FUNCT_1:51
        .= pk by FINSEQ_1:def 16;
      set args = the_arity_of o;
      let x be Element of D;
A33:  dom a = dom args by MSATERM:22;
A34:  1<=k+1 by NAT_1:11;
      then
A35:  k+1 in dom p by A4,A6,A11,A25,FINSEQ_3:25;
A36:  len pk1 = k+1 by A4,A6,A11,A25,FINSEQ_1:59;
      then
A37:  k+1 in dom pk1 by A34,FINSEQ_3:25;
      p1 = <*p.(k+1)*> by A4,A6,A11,A19,A22,A29,RFINSEQ:def 1
        .= <*p/.(k+1)*> by A35,PARTFUN1:def 6
        .= <*(p|(k+1))/.(k+1)*> by A37,FINSEQ_4:70
        .= <*pk1.(k+1)*> by A37,PARTFUN1:def 6;
      then pk1 = pk^p1 by A36,A32,RFINSEQ:7;
      then
A38:  t29 = t2 by TREES_9:3;
      1<=i+1 & i+1<=len a by A30,NAT_1:11,13;
      then
A39:  i+1 in dom args by A33,FINSEQ_3:25;
      then reconsider rs1 = args.(i+1) as SortSymbol of S by DTCONSTR:2;
      set e1 = [o,rs1];
A40:  (the Arity of S).o = the_arity_of o by MSUALG_1:def 1;
      then [o,rs1] in InducedEdges S by A39,Def1;
      then reconsider E9 = the carrier' of G as non empty set;
      reconsider e19 = e1 as Element of E9 by A39,A40,Def1;
      reconsider x9 = x as FinSequence of E9 by FINSEQ_1:def 11;
      reconsider y = <*e19*>^x9 as Element of D by FINSEQ_1:def 11;
      take y;
      take t1, t2;
      thus t1 = t|(p|k) & t2 = t|(p|(k+1));
      take o, rs1, x;
      thus x = x & y = <*[o,rs1]*>^x;
      thus [o,the carrier of S] = t1.{} by A27;
      pk9|1 = <*i*> by A31,A29,FINSEQ_1:41;
      then t29 = a.(i+1) by A28,A30,TREES_4:def 4;
      hence rs1 = the_sort_of t2 by A33,A39,A38,MSATERM:23;
      thus thesis by A39,A40,Def1;
    end;
    consider C being FinSequence of D such that
A41: len C = n1 & (C.1 = C19 or n1 = 0) & for k being Nat
    st 1 <= k & k < n1 holds P[k,C.k,C.(k+1)] from RECDEF_1:sch 4(A17);
    defpred Z[Nat] means 1<=$1 & $1<=n implies ex Ck being directed Chain of G
    , t1 being Term of S, X st Ck = C.$1 & len Ck = $1 & t1 = t|(p|$1) & (
    vertex-seq Ck).(len Ck +1) = v & (vertex-seq Ck).1 = the_sort_of t1;
A42: for i be Nat st Z[i] holds Z[i+1]
    proof
      let k;
      assume
A43:  1<=k & k<=n implies ex Ck being directed Chain of G, t1 being
Term of S, X st Ck = C.k & len Ck = k & t1 = t|(p|k) & (vertex-seq Ck).(len Ck
      +1) = v & (vertex-seq Ck).1 = the_sort_of t1;
A44:  k<=k+1 by NAT_1:11;
      assume that
      1<=k+1 and
A45:  k+1<=n;
A46:  k<n by A45,NAT_1:13;
      per cases;
      suppose
A47:    k=0;
        reconsider C1 = <*e1*> as directed Chain of G by A16,MSSCYC_1:5;
        reconsider p1 = p|(0+1) as Node of t by A5,A10,MSSCYC_1:19;
        reconsider t2 = t9|p1 as Term of S, X by MSATERM:29;
        take C1, t2;
        thus C1 = C.(k+1) by A1,A41,A47;
        thus len C1 = k+1 by A47,FINSEQ_1:39;
        thus t2 = t|(p|(k+1)) by A47;
        reconsider p9=p as PartFunc of NAT, NAT;
A48:    vertex-seq C1 = <*(the Source of G).e1, (the Target of G).e1*> by A16,
MSSCYC_1:7;
        (vertex-seq C1).(len C1 +1) = (vertex-seq C1).(1+1) by FINSEQ_1:39
          .= (the Target of G).e1 by A48
          .= (the ResultSort of S).e1`1 by A16,Def3
          .= (the ResultSort of S).o
          .= the_result_sort_of o by MSUALG_1:def 2
          .= the_sort_of t9 by A7,MSATERM:17
          .= v by A2,MSATERM:def 5;
        hence (vertex-seq C1).(len C1 +1) = v;
        p|1 = <*p9.1*> by A1,A4,A6,A11,CARD_1:27,FINSEQ_5:20
          .= <*p.1*>
          .= <*i*> by A13,FINSEQ_1:41;
        then
A50:    t2 = a.(i+1) by A8,A12,TREES_4:def 4;
        (vertex-seq C1).1 = (the Source of G).e1 by A48
          .= e1`2 by A16,Def2
          .= rs1
          .= the_sort_of t2 by A9,A14,A50,MSATERM:23;
        hence thesis;
      end;
      suppose
A51:    k<>0;
        consider t1, t2 being Term of S, X such that
A52:    t1 = t|(p|k) and
A53:    t2 = t|(p|(k+1)) and
A54:    ex o being OperSymbol of S,rs1 being SortSymbol of S,Ck being
Element of D st Ck = C.k & C.(k+1) = <*[o,rs1]*>^Ck & [o,the carrier of S] = t1
.{}& rs1 = the_sort_of t2 & [o,rs1] in the carrier' of G by A41,A46,A51,
NAT_1:14;
        consider o being OperSymbol of S, rs1 being SortSymbol of S, Ck9 being
        Element of D such that
A55:    Ck9 = C.k & C.(k+1) = <*[o,rs1]*>^Ck9 and
A56:    [o,the carrier of S] = t1.{} and
A57:    rs1 = the_sort_of t2 and
A58:    [o,rs1] in the carrier' of G by A54;
A59:    G is non void by A58;
        reconsider C1 = <*[o,rs1]*> as directed Chain of G by A58,MSSCYC_1:5;
        set e1 = [o,rs1];
A60:    vertex-seq C1 = <*(the Source of G).e1, (the Target of G).e1*> by A58,
MSSCYC_1:7;
        consider Ck being directed Chain of G, t19 being Term of S, X such
        that
A61:    Ck = C.k and
A62:    len Ck = k and
A63:    t19 = t|(p|k) and
A64:    (vertex-seq Ck).(len Ck +1)=v and
A65:    (vertex-seq Ck).1=the_sort_of t19 by A43,A45,A44,A51,NAT_1:14
,XXREAL_0:2;
        (vertex-seq C1).(len C1 +1) = (vertex-seq C1).(1+1) by FINSEQ_1:39
          .= (the Target of G).e1 by A60
          .= (the ResultSort of S).e1`1 by A58,Def3
          .= (the ResultSort of S).o
          .= the_result_sort_of o by MSUALG_1:def 2
          .= the_sort_of t1 by A56,MSATERM:17;
        then reconsider
        d = C1^Ck as directed Chain of G by A51,A62,A63,A65,A52,A59,CARD_1:27
,MSSCYC_1:15;
        take d, t2;
        thus d = C.(k+1) by A61,A55;
        thus len d = len C1 + k by A62,FINSEQ_1:22
          .= k+1 by FINSEQ_1:39;
        thus t2 = t|(p|(k+1)) by A53;
        thus (vertex-seq d).(len d +1) = v by A51,A62,A64,A59,CARD_1:27
,MSSCYC_1:16;
        (vertex-seq C1).1 = (the Source of G).e1 by A60
          .= e1`2 by A58,Def2
          .= the_sort_of t2 by A57;
        hence thesis by A51,A62,A59,CARD_1:27,MSSCYC_1:16;
      end;
    end;
A66: Z[ 0 ];
    for k holds Z[k] from NAT_1:sch 2(A66, A42);
    then
    ex c being directed Chain of G, t1 being Term of S, X st c = C.n & len
    c = n & t1 = t|(p|n) & (vertex-seq c).(len c +1) = v & ( vertex-seq c).1 =
    the_sort_of t1 by A1;
    hence thesis;
  end;
  given c being directed Chain of InducedGraph S such that
A67: len c = n and
A68: (vertex-seq c).(len c +1) = v;
  set EG = the carrier' of G;
  reconsider n1=n as Element of NAT by ORDINAL1:def 12;
  deffunc F(object) = X.$1;
  set TG = the Target of G;
  set SG = the Source of G;
  set D = S-Terms X;
  set cS = the carrier of S;
A69: for e being object st e in cS holds F(e) <> {};
  consider cX being ManySortedSet of cS such that
A70: for e being object st e in cS holds cX.e in F(e) from PBOOLE:sch 1 (
  A69);
  defpred P[Nat, set, set] means ex o being OperSymbol of S, rs1 being
SortSymbol of S, Ck, Ck1 being Term of S, X, a being ArgumentSeq of Sym(o,X) st
  Ck = $2 & $3 = Ck1 & c.($1+1) = [o, rs1] & Ck1 = [o,cS]-tree a & (for i being
  Nat st i in dom a ex t being Term of S,X st t = a.i & the_sort_of t = (
the_arity_of o).i & (the_sort_of t = rs1 & the_sort_of Ck = rs1 implies t = Ck)
  & (the_sort_of t <> rs1 or the_sort_of Ck <> rs1 implies t = root-tree [cX.(
  the_sort_of t), the_sort_of t]));
A71: c is FinSequence of the carrier' of InducedGraph S by MSSCYC_1:def 1;
A72: 1 in dom c by A1,A67,FINSEQ_3:25;
  then reconsider EG as non empty set by A71;
  c.1 in InducedEdges S by A71,A72,DTCONSTR:2;
  then consider o, rs1 being set such that
A73: c.1 = [o, rs1] and
A74: o in the carrier' of S and
A75: rs1 in the carrier of S and
A76: ex n being Nat, args being Element of (the carrier of S)* st (the
  Arity of S).o = args & n in dom args & args.n = rs1 by Def1;
  reconsider rs1 as SortSymbol of S by A75;
  reconsider o as OperSymbol of S by A74;
  deffunc F(Nat) = root-tree [cX.((the_arity_of o).$1),(the_arity_of o).$1];
  consider a being FinSequence such that
A77: len a = len the_arity_of o & for k be Nat st k in dom a holds a.k
  = F(k) from FINSEQ_1:sch 2;
A78: dom a = Seg len a by FINSEQ_1:def 3;
A79: for i being Nat st i in dom a ex t being Term of S,X st t = a.i &
  the_sort_of t = (the_arity_of o).i
  proof
    let i be Nat;
    assume
A80: i in dom a;
    set s = (the_arity_of o).i;
    dom the_arity_of o = Seg len the_arity_of o by FINSEQ_1:def 3;
    then reconsider s as SortSymbol of S by A77,A78,A80,DTCONSTR:2;
    set x = cX.((the_arity_of o).i);
    reconsider x as Element of X.s by A70;
    reconsider t = root-tree [x,s] as Term of S,X by MSATERM:4;
    take t;
    thus t = a.i by A77,A80;
    thus thesis by MSATERM:14;
  end;
A81: dom a = Seg len the_arity_of o by A77,FINSEQ_1:def 3;
  reconsider a as ArgumentSeq of Sym(o,X) by A77,A79,MSATERM:24;
  set C1 = [o,the carrier of S]-tree a;
  reconsider C1 as Term of S, X by MSATERM:1;
A82: for k being Nat st 1 <= k & k < n1 for x being Element of D
  ex y being Element of D st P[k,x,y]
  proof
    let k be Nat;
    assume that
    1 <= k and
A83: k < n1;
A84: 1<=k+1 by NAT_1:11;
    k+1<=n by A83,NAT_1:13;
    then k+1 in dom c by A67,A84,FINSEQ_3:25;
    then reconsider ck1 = c.(k+1) as Element of EG by A71,DTCONSTR:2;
    let x be Element of D;
    consider o, rs1 being set such that
A85: ck1 = [o,rs1] and
A86: o in the carrier' of S and
A87: rs1 in cS and
    ex n being Nat, args being Element of (the carrier of S)* st (the
    Arity of S).o = args & n in dom args & args.n = rs1 by Def1;
    reconsider rs1 as SortSymbol of S by A87;
    reconsider o as OperSymbol of S by A86;
    set DA = dom the_arity_of o;
    set ar = the_arity_of o;
    defpred P[object, object] means
(ar.$1 = rs1 & the_sort_of x = rs1 implies $2 =
x) & (ar.$1 <> rs1 or the_sort_of x <> rs1 implies $2 = root-tree [cX.(ar.$1),
    ar.$1]);
A88: for e being object st e in DA ex u being object st u in D & P[e,u]
    proof
      let e be object;
      assume
A89:  e in DA;
      per cases;
      suppose
A90:    ar.e = rs1 & the_sort_of x = rs1;
        take x;
        thus thesis by A90;
      end;
      suppose
A91:    ar.e <> rs1 or the_sort_of x <> rs1;
        reconsider s = (the_arity_of o).e as SortSymbol of S by A89,DTCONSTR:2;
        reconsider x = cX.((the_arity_of o).e) as Element of X.s by A70;
        reconsider t = root-tree [x,s] as Term of S,X by MSATERM:4;
        take t;
        thus thesis by A91;
      end;
    end;
    consider a being Function of DA,D such that
A92: for e being object st e in DA holds P[e,a.e] from FUNCT_2:sch 1 (
    A88);
    DA = Seg len ar by FINSEQ_1:def 3;
    then reconsider a as FinSequence of D by FINSEQ_2:25;
A93: dom a = DA by FUNCT_2:def 1;
    now
      let i be Nat;
      assume
A94:  i in dom a;
      then reconsider t = a.i as Term of S,X by DTCONSTR:2;
      take t;
      thus t = a.i;
      per cases;
      suppose
        ar.i = rs1 & the_sort_of x = rs1;
        hence the_sort_of t = ar.i by A92,A93,A94;
      end;
      suppose
A95:    ar.i <> rs1 or the_sort_of x <> rs1;
        reconsider s = (the_arity_of o).i as SortSymbol of S by A93,A94,
DTCONSTR:2;
A96:    cX.((the_arity_of o).i) is Element of X.s by A70;
        t = root-tree [cX.(ar.i),ar.i] by A92,A93,A94,A95;
        hence the_sort_of t = ar.i by A96,MSATERM:14;
      end;
    end;
    then reconsider a as ArgumentSeq of Sym(o,X) by A93,MSATERM:24;
    reconsider y = [o,cS]-tree a as Term of S,X by MSATERM:1;
    take y, o, rs1, x, y, a;
    thus x = x & y = y;
    thus c.(k+1) = [o, rs1] by A85;
    thus y = [o,cS]-tree a;
    let i be Nat;
    assume
A97: i in dom a;
    then reconsider t = a.i as Term of S,X by DTCONSTR:2;
    take t;
    thus t = a.i;
    thus the_sort_of t = (the_arity_of o).i by A97,MSATERM:23;
    hence thesis by A92,A93,A97;
  end;
  consider C being FinSequence of D such that
A98: len C = n1 & (C.1 = C1 or n1 = 0) & for k be Nat st 1
  <= k & k < n1 holds P[k,C.k,C.(k+1)] from RECDEF_1:sch 4(A82);
  defpred P[Nat] means 1<=$1 & $1<=n implies ex C0 being Term of S, X, o being
  OperSymbol of S st C0 = C.$1 & o = (c.$1)`1 & the_sort_of C0 =
  the_result_sort_of o & height dom C0 = $1;
A99: G is non void by A72;
A100: for k be Nat st P[k] holds P[k+1]
  proof
    let k;
    assume
A101: 1<=k & k<=n implies ex Ck being Term of S, X, o being OperSymbol
of S st Ck = C.k & o = (c.k)`1 & the_sort_of Ck = the_result_sort_of o & height
    dom Ck = k;
    assume that
A102: 1<=k+1 and
A103: k+1<=n;
A104: k<n by A103,NAT_1:13;
A105: k<=k+1 by NAT_1:11;
    then
A106: k<=n by A103,XXREAL_0:2;
    per cases;
    suppose
A107: k = 0;
      take C1, o;
      thus C1 = C.(k+1) by A1,A98,A107;
      thus o = (c.(k+1))`1 by A73,A107;
      reconsider w = doms a as FinTree-yielding FinSequence;
A108: dom doms a = dom a by TREES_3:37;
      C1.{} = [o, cS] by TREES_4:def 4;
      hence the_sort_of C1 = the_result_sort_of o by MSATERM:17;
      consider i being Nat, args being Element of (the carrier of S)* such
      that
A109: (the Arity of S).o = args and
A110: i in dom args and
      args.i = rs1 by A76;
A111: dom args = Seg len args by FINSEQ_1:def 3;
A112: args = the_arity_of o by A109,MSUALG_1:def 1;
      then reconsider t = a.i as Term of S, X by A77,A78,A110,A111,MSATERM:22;
      (doms a).i = dom t by A77,A78,A110,A112,A111,FUNCT_6:22;
      then
A113: dom C1 = tree doms a & dom t in rng w by A77,A78,A108,A110,A112,A111,
FUNCT_1:def 3,TREES_4:10;
      reconsider dt = dom t as finite Tree;
A114: a.i = root-tree [cX.((the_arity_of o).i),(the_arity_of o).i] by A77,A81
,A110,A112,A111;
A115: now
        let t9 be finite Tree;
        assume t9 in rng w;
        then consider j being Nat such that
A116:   j in dom w and
A117:   w.j = t9 by FINSEQ_2:10;
        reconsider t99 = a.j as Term of S, X by A108,A116,MSATERM:22;
        a.j = root-tree [cX.((the_arity_of o).j),(the_arity_of o).j] by A77
,A108,A116;
        then
A118:   dom t99 = elementary_tree 0 by TREES_4:3;
        w.j = dom t99 by A108,A116,FUNCT_6:22;
        hence height t9 <= height dt by A114,A117,A118,TREES_4:3;
      end;
      dom t = elementary_tree 0 by A114,TREES_4:3;
      hence thesis by A107,A113,A115,TREES_1:42,TREES_3:79;
    end;
    suppose
A119: k <> 0;
      then
A120: 1<=k by NAT_1:14;
      then k in dom c by A67,A106,FINSEQ_3:25;
      then reconsider ck = c.k as Element of EG by A71,DTCONSTR:2;
      consider Ck being Term of S, X, o being OperSymbol of S such that
A121: Ck = C.k and
A122: o = (c.k)`1 & the_sort_of Ck = the_result_sort_of o and
A123: height dom Ck = k by A101,A103,A105,A119,NAT_1:14,XXREAL_0:2;
      reconsider kk=k as Element of NAT by ORDINAL1:def 12;
      consider o1 being OperSymbol of S, rs1 being SortSymbol of S, Ck9, Ck1
      being Term of S, X, a being ArgumentSeq of Sym(o1,X) such that
A124: Ck9 = C.k and
A125: C.(kk+1) = Ck1 and
A126: c.(k+1) = [o1, rs1] and
A127: Ck1 = [o1,cS]-tree a and
A128: for i being Nat st i in dom a ex t being Term of S,X st t = a.
i & the_sort_of t = (the_arity_of o1).i & (the_sort_of t = rs1 & the_sort_of
Ck9 = rs1 implies t = Ck9) & (the_sort_of t <> rs1 or the_sort_of Ck9 <> rs1
implies t = root-tree [cX.(the_sort_of t), the_sort_of t]) by A98,A104,A119,
NAT_1:14;
      set ck1 = c.(kk+1);
A129: k+1 in dom c by A67,A102,A103,FINSEQ_3:25;
      then ck1 in EG by A71,DTCONSTR:2;
      then consider o9, rs19 being set such that
A130: ck1=[o9, rs19] and
A131: o9 in the carrier' of S and
      rs19 in the carrier of S and
A132: ex n being Nat, args being Element of (the carrier of S)* st (
      the Arity of S).o9 = args & n in dom args & args.n = rs19 by Def1;
A133: o1 = o9 by A126,A130,XTUPLE_0:1;
      take Ck1, o1;
      thus Ck1 = C.(k+1) by A125;
      thus o1 = (c.(k+1))`1 by A126;
      Ck1.{} = [o1,cS] by A127,TREES_4:def 4;
      hence the_sort_of Ck1 = the_result_sort_of o1 by MSATERM:17;
A134: dom Ck1 = tree doms a by A127,TREES_4:10;
      reconsider ck1 as Element of EG by A71,A129,DTCONSTR:2;
      reconsider w = doms a as FinTree-yielding FinSequence;
A135: len a = len the_arity_of o1 & dom a = Seg len a by FINSEQ_1:def 3
,MSATERM:22;
      rs1 = rs19 by A126,A130,XTUPLE_0:1;
      then consider
      i being Nat, args being Element of (the carrier of S)* such
      that
A136: (the Arity of S).o9 = args and
A137: i in dom args and
A138: args.i = rs1 by A132;
      reconsider o9 as OperSymbol of S by A131;
A139: dom args = Seg len args by FINSEQ_1:def 3;
A140: args = the_arity_of o9 by A136,MSUALG_1:def 1;
      then consider t being Term of S, X such that
A141: t = a.i and
A142: the_sort_of t = (the_arity_of o1).i &( the_sort_of t = rs1 &
      the_sort_of Ck9 = rs1 implies t = Ck9) and
      the_sort_of t <> rs1 or the_sort_of Ck9 <> rs1 implies t = root-tree
      [cX.(the_sort_of t), the_sort_of t] by A128,A135,A133,A137,A139;
      reconsider dt = dom t as finite Tree;
A143: dom doms a = dom a by TREES_3:37;
A144: now
        let t9 be finite Tree;
        assume t9 in rng w;
        then consider j being Nat such that
A145:   j in dom w and
A146:   w.j = t9 by FINSEQ_2:10;
        consider t99 being Term of S, X such that
A147:   t99 = a.j and
        the_sort_of t99 = (the_arity_of o1).j and
A148:   the_sort_of t99 = rs1 & the_sort_of Ck9 = rs1 implies t99 = Ck9 and
A149:   the_sort_of t99 <> rs1 or the_sort_of Ck9 <> rs1 implies t99
        = root-tree [cX.(the_sort_of t99), the_sort_of t99] by A128,A143,A145;
A150:   w.j = dom t99 by A143,A145,A147,FUNCT_6:22;
        per cases;
        suppose
          the_sort_of t99 = rs1 & the_sort_of Ck9 = rs1;
          hence height t9 <= height dt by A143,A133,A136,A138,A142,A145,A146
,A147,A148,FUNCT_6:22,MSUALG_1:def 1;
        end;
        suppose
          the_sort_of t99 <> rs1 or the_sort_of Ck9 <> rs1;
          hence height t9 <= height dt by A146,A149,A150,TREES_1:42,TREES_4:3;
        end;
      end;
      (doms a).i = dom t by A135,A133,A137,A140,A139,A141,FUNCT_6:22;
      then
A151: dom t in rng w by A143,A135,A133,A137,A140,A139,FUNCT_1:def 3;
      the_sort_of Ck = (the ResultSort of S).(ck`1) by A122,MSUALG_1:def 2
        .= TG.ck by Def3
        .= (vertex-seq c).(kk+1) by A67,A99,A106,A120,CARD_1:27,MSSCYC_1:11
        .= SG.ck1 by A67,A99,A102,A103,CARD_1:27,MSSCYC_1:11
        .= ck1`2 by Def2
        .= rs1 by A126;
      hence thesis by A121,A123,A124,A134,A133,A136,A138,A142,A151,A144,
MSUALG_1:def 1,TREES_3:79;
    end;
  end;
  set cn = c.len c;
  n in dom c by A1,A67,CARD_1:27,FINSEQ_5:6;
  then
A152: cn in InducedEdges S by A67,A71,DTCONSTR:2;
A153: P[ 0 ];
  for k holds P[k] from NAT_1:sch 2(A153, A100);
  then consider Cn being Term of S, X, o being OperSymbol of S such that
  Cn = C.n and
A154: o = (c.n)`1 and
A155: the_sort_of Cn = the_result_sort_of o and
A156: height dom Cn = n by A1;
  G is non void by A72;
  then (vertex-seq c).(len c +1) = (the Target of G).(c.len c) by A1,A67,
CARD_1:27,MSSCYC_1:14
    .= (the ResultSort of S).cn`1 by A152,Def3
    .= the_result_sort_of o by A67,A154,MSUALG_1:def 2;
  then reconsider Cn as Element of (the Sorts of FreeMSA X).v by A2,A68,A155,
MSATERM:def 5;
  take Cn;
  thus thesis by A156,MSAFREE2:def 14;
end;
