theorem Th53:
  p is x-context_including iff o-term p is context of x
  proof
    set I = {[x,s]}; set k = p;
    reconsider J = [o,the carrier of S] as set;
    s in the carrier of S; then s <> the carrier of S;
    then [o,the carrier of S] <> [x,s] by XTUPLE_0:1;
    then [o,the carrier of S] nin I by TARSKI:def 1;
    then IFIN(J,I,{{}},{}) = {} by MATRIX_7:def 1;
    then
A5: (o-term k)"I = {} \/ union {<*i*>^^((k.(i+1))"I): i < len k} by Th80;
    thus k is x-context_including implies o-term k is context of x
    proof
      given i such that
A6:   i in dom k & k.i is context of x &
      for j for t being Element of Free(S,X) st j in dom k & j <> i & t = k.j
      holds t is x-omitting;
      reconsider C = k.i as context of x by A6;
      card Coim(C,[x,s]) = 1 by CONTEXT;
      then consider a such that
A9:   C"I = {a} by CARD_2:42;
      a in C"I by A9,TARSKI:def 1;
      then reconsider a as FinSequence;
      consider j such that
A7:   i = 1+j by A6,FINSEQ_3:25,NAT_1:10;
      1+j <= len k by A6,A7,FINSEQ_3:25;
      then
A8:   j < len k by NAT_1:13;
      union {<*i*>^^((k.(i+1))"I) where i: i < len k} = {<*j*>^a}
      proof
        thus union {<*i*>^^((k.(i+1))"I) where i: i < len k} c= {<*j*>^a}
        proof
          let b; assume b in union {<*i*>^^((k.(i+1))"I) where i: i < len k};
          then consider J such that
B1:       b in J in {<*i*>^^((k.(i+1))"I) where i: i < len k} by TARSKI:def 4;
          consider n being Nat such that
B2:       J = <*n*>^^((k.(n+1))"I) & n < len k by B1;
B4:       1 <= n+1 <= len k by B2,NAT_1:11,13;
          then n+1 in dom k by FINSEQ_3:25;
          then reconsider t = k.(n+1) as Element of Free(S,X) by FUNCT_1:102;
          per cases;
          suppose n = j;
            hence thesis by B1,B2,A9,A7,Th6;
          end;
          suppose n <> j;
            then n+1 <> j+1;
            then t is x-omitting by A6,A7,B4,FINSEQ_3:25;
            hence thesis by B1,B2;
          end;
        end;
        let b; assume b in {<*j*>^a};
        then b in <*j*>^^(C"I) in {<*i*>^^((k.(i+1))"I) where i: i < len k}
        by A7,A8,A9,Th6;
        hence thesis by TARSKI:def 4;
      end;
      then card Coim(o-term k,[x,s]) = 1 by A5,CARD_1:30;
      hence thesis by CONTEXT;
    end;
    assume o-term k is context of x;
    then card Coim(o-term k, [x,s]) = 1 by CONTEXT;
    then consider a such that
D1: Coim(o-term k, [x,s]) = {a} by CARD_2:42;
    a in (o-term k)"I by D1,TARSKI:def 1;
    then consider J such that
D2: a in J in {<*i*>^^((k.(i+1))"I): i < len k} by A5,TARSKI:def 4;
    consider i such that
D3: J = <*i*>^^((k.(i+1))"I) & i < len k by D2;
    consider p being Element of (k.(i+1))"I such that
D4: a = <*i*>^p & p in (k.(i+1))"I by D2,D3;
    take n = i+1;
    1 <= n <= len k by D3,NAT_1:11,13;
    hence n in dom k by FINSEQ_3:25;
    then reconsider kn = k.n as Element of Free(S,X) by FUNCT_1:102;
D6: {p} c= kn"I by D4,ZFMISC_1:31;
    J c= {a} = <*i*>^^{p} by A5,D1,D2,D4,Th6,ZFMISC_1:74;
    then kn"I c= {p} by D3,Th18;
    then kn"I = {p} by D6,XBOOLE_0:def 10;
    then card Coim(kn,[x,s]) = 1 by CARD_1:30;
    hence k.n is context of x by CONTEXT;
    let j;
    let t be Element of Free(S,X); assume
D7: j in dom k & j <> n & t = k.j;
    assume Coim(t,[x,s]) <> {};
    then consider c being object such that
D8: c in Coim(t,[x,s]) by XBOOLE_0:7;
    reconsider c as Node of t by D8,FUNCT_1:def 7;
    consider m being Nat such that
D9: j = 1+m by D7,FINSEQ_3:25,NAT_1:10;
    1+m <= len k by D7,D9,FINSEQ_3:25;
    then m < len k by NAT_1:13;
    then <*m*>^c in <*m*>^^(t"I) in
    {<*i*>^^((k.(i+1))"I) where i: i < len k} by D7,D8,D9;
    then <*m*>^c in {a} by A5,D1,TARSKI:def 4;
    then <*m*>^c = <*i*>^p by D4,TARSKI:def 1;
    then m = (<*m*>^c).1 = i by FINSEQ_1:41;
    hence contradiction by D7,D9;
  end;
