reserve a,x,y for object, A,B for set,
  l,m,n for Nat;
reserve X,Y for set, x for object,
  p,q for Function-yielding FinSequence,
  f,g,h for Function;
reserve m,n,k for Nat, R for Relation;
reserve i,j for Nat;
reserve F for Function,
  e,x,y,z for object;

theorem
  for f,g be FinSequence of INT,m,n be Nat st 1<=n & n <= len
f & 1<=m & m <= len f & g=f+*(m,f/.n) +*(n,f/.m) holds f.m=g.n & f.n=g.m & (for
  k be set st k<>m & k<>n & k in dom f holds f.k=g.k) & 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: g=f+*(m,f/.n) +*(n,f/.m);
A4: dom (f+*(m,f/.n))=dom f by Th29;
A5: n in dom f by A1,FINSEQ_3:25;
  hence
A6: g.n=f/.m by A3,A4,Th30
    .=f.m by A2,FINSEQ_4:15;
A7: m in dom f by A2,FINSEQ_3:25;
  thus
A8: now
    per cases;
    suppose
      m=n;
      hence g.m=f.n by A6;
    end;
    suppose
      m<>n;
      hence g.m=(f+*(m,f/.n)).m by A3,Th31
        .=f/.n by A7,Th30
        .=f.n by A1,FINSEQ_4:15;
    end;
  end;
A9: now
    let k be set;
    assume that
A10: k<>m and
A11: k<>n and
    k in dom f;
    thus g.k=(f+*(m,f/.n)).k by A3,A11,Th31
      .=f.k by A10,Th31;
  end;
  dom g =dom f by A3,A4,Th29;
  hence thesis by A5,A7,A6,A8,A9,RFINSEQ:28;
end;
