reserve x for Int_position,
  n,p0 for Nat;

theorem Th3:
  for f,g be FinSequence of INT,m,n be Nat st 1<=n & n
  <= len f & 1<=m & m <= len f & len f=len g & f.m=g.n & f.n=g.m & (for k be
  Nat st k<>m & k<>n & 1<=k & k <= len f holds f.k=g.k) holds f,g
  are_fiberwise_equipotent
proof
  let f,g be FinSequence of INT,m,n be Nat;
  assume that
A1: 1<=n & n <= len f and
A2: 1<=m & m <= len f and
A3: len f=len g and
A4: f.m=g.n & f.n=g.m and
A5: for k be Nat st k<>m & k<>n & 1<=k & k <= len f holds f.k
  =g.k;
A6: m in Seg (len f) by A2,FINSEQ_1:1;
A7: Seg (len f) = dom f by FINSEQ_1:def 3;
A8: now
    let k be set;
    assume that
A9: k<>m & k<>n and
A10: k in dom f;
    reconsider i=k as Nat by A10;
    1 <= i & i <= len f by A7,A10,FINSEQ_1:1;
    hence f.k=g.k by A5,A9;
  end;
  n in dom f & dom f=dom g by A1,A3,A7,FINSEQ_1:1,def 3;
  hence thesis by A4,A7,A6,A8,RFINSEQ:28;
end;
