reserve x,y for set,
  n,m for Nat,
  r,s for Real;
reserve f, g for Function;

theorem
  for f,g be FinSequence,m,n,j be Nat st f,g are_fiberwise_equipotent &
  m<=n & n <= len f & (for i be Nat st 1<=i & i<=m holds f.i=g.i) &
  (for i be Nat st n<i & i<=len f holds f.i=g.i) & m<j & j<=n
  ex k be Nat st m<k & k<=n & f.j=g.k
proof
  let f,g be FinSequence,m,n,j be Nat;
  assume
A1: f,g are_fiberwise_equipotent;
  assume that
A2: m<=n and
A3: n <= len f;
  assume
A4: for i be Nat st 1<=i & i<=m holds f.i=g.i;
  reconsider m3=len f-n as Element of NAT by A3,INT_1:3,XREAL_1:48;
  len g=n+m3 by A1,Th3;
  then consider s2,r2 be FinSequence such that
A5: len s2 = n and
A6: len r2 = m3 and
A7: g = s2^r2 by FINSEQ_2:22;
A8: len f=n+m3;
  then consider s1,r1 be FinSequence such that
A9: len s1 = n and
A10: len r1 = m3 and
A11: f = s1^r1 by FINSEQ_2:22;
A12: dom r1 = Seg m3 by A10,FINSEQ_1:def 3;
  assume
A13: for i be Nat st n<i & i<=len f holds f.i=g.i;
  now
    let i be Nat;
    reconsider a = i as Nat;
A14: n < n+1 by XREAL_1:29;
    assume
A15: i in dom r1;
    then
A16: i in dom r2 by A6,A12,FINSEQ_1:def 3;
    1 <= i by A12,A15,FINSEQ_1:1;
    then n+1 <= n+i by XREAL_1:6;
    then
A17: n< i+n by A14,XXREAL_0:2;
    i <= m3 by A12,A15,FINSEQ_1:1;
    then
A18: len s1+i <= len f by A8,A9,XREAL_1:6;
    thus r1.i=f.(len s1+i) by A11,A15,FINSEQ_1:def 7
      .=g.(len s2+a) by A13,A9,A5,A17,A18
      .=r2.i by A7,A16,FINSEQ_1:def 7;
  end;
  then r1=r2 by A10,A6,FINSEQ_2:9;
  then
A19: s1,s2 are_fiberwise_equipotent by A1,A11,A7,Th1;
  reconsider m2=n-m as Element of NAT by A2,INT_1:3,XREAL_1:48;
A20: m+1 > m by XREAL_1:29;
  len s2=m+m2 by A5;
  then consider p2,q2 be FinSequence such that
A21: len p2 = m and
A22: len q2 = m2 and
A23: s2 = p2^q2 by FINSEQ_2:22;
A24: Seg m = dom p2 by A21,FINSEQ_1:def 3;
  len s1=m+m2 by A9;
  then consider p1,q1 be FinSequence such that
A25: len p1 = m and
A26: len q1 = m2 and
A27: s1 = p1^q1 by FINSEQ_2:22;
A28: f=p1^(q1^r1) by A11,A27,FINSEQ_1:32;
A29: dom p1 = Seg m by A25,FINSEQ_1:def 3;
A30: g=p2^(q2^r2) by A7,A23,FINSEQ_1:32;
  now
    let i be Nat;
    reconsider a = i as Nat;
    assume
A31: i in dom p1;
    then
A32: 1<= i & i <= m by A29,FINSEQ_1:1;
    thus p1.i=f.i by A28,A31,FINSEQ_1:def 7
      .=g.a by A4,A32
      .=p2.i by A30,A24,A29,A31,FINSEQ_1:def 7;
  end;
  then p1=p2 by A25,A21,FINSEQ_2:9;
  then
A33: q1,q2 are_fiberwise_equipotent by A27,A23,A19,Th31;
  assume that
A34: m<j and
A35: j<=n;
  j-len p1 > 0 by A34,A25,XREAL_1:50;
  then reconsider x=j-len p1 as Element of NAT by INT_1:3;
A36: x <= n-len p1 by A35,XREAL_1:9;
A37: Seg m2 = dom q2 by A22,FINSEQ_1:def 3;
A38: 1+ 0<= x by A34,A25,INT_1:7,XREAL_1:50;
  then x in dom q2 by A25,A37,A36;
  then consider y be set such that
A39: y in dom q2 and
A40: q1.x=q2.y by A33,Th30;
  reconsider y as Nat by A39;
A41: len p2 + y in dom s2 by A23,A39,FINSEQ_1:28;
  reconsider k=len p2+y as Nat;
  take k;
  1<=y by A37,A39,FINSEQ_1:1;
  then k>=len p2+1 by XREAL_1:6;
  hence m<k by A21,A20,XXREAL_0:2;
  y<=m2 by A37,A39,FINSEQ_1:1;
  then k <= m2+len p2 by XREAL_1:6;
  hence k<=n by A21;
  Seg m2 = dom q1 by A26,FINSEQ_1:def 3;
  then
A42: x in dom q1 by A25,A38,A36;
  then len p1 + x in dom s1 by A27,FINSEQ_1:28;
  hence f.j=s1.(len p1+x) by A11,FINSEQ_1:def 7
    .=q2.y by A27,A40,A42,FINSEQ_1:def 7
    .=s2.(len p2+y) by A23,A39,FINSEQ_1:def 7
    .=g.k by A7,A41,FINSEQ_1:def 7;
end;
