reserve n,m for Nat;

theorem
  for f being real-valued FinSequence st len f >=1 holds
  max_p(sort_d f)=1 & min_p(sort_a f)=1 &
  (sort_d f).1=max f & (sort_a f).1=min f
proof
  let f be real-valued FinSequence;
  assume
A1: len f>=1;
A2: len (sort_d f)=len f by Th28;
  then 1 in Seg len (sort_d f) by A1,FINSEQ_1:1;
  then
A3: 1 in dom (sort_d f) by FINSEQ_1:def 3;
A4: for i being Nat st i in dom (sort_d f)  holds (sort_d f).i<=(sort_d f).1
  proof
    let i be Nat;
    assume
A5: i in dom (sort_d f);
    set r1=(sort_d f).i;
    set r2=(sort_d f).1;
    i in Seg len (sort_d f) by A5,FINSEQ_1:def 3;
    then
A7: 1<=i by FINSEQ_1:1;
    now
      per cases;
      case
        1=i;
        hence thesis;
      end;
      case
        1<>i;
        then 1<i by A7,XXREAL_0:1;
        hence thesis by A3,A5,RFINSEQ:19;
      end;
    end;
    hence thesis;
  end;
A8: len (sort_a f)=len f by Th29;
  then
A9: 1 in dom (sort_a f) by A1,FINSEQ_3:25;
A10: for i being Nat st i in dom (sort_a f) holds (sort_a f).i>=(sort_a f).1
  proof
    let i be Nat;
    assume that
A11: i in dom (sort_a f);
     set r1=(sort_a f).i;
     set r2=(sort_a f).1;
A13: 1<=i by A11,FINSEQ_3:25;
    per cases;
    suppose
      1=i;
      hence thesis;
    end;
    suppose
      1<>i;
      then 1<i by A13,XXREAL_0:1;
      hence thesis by A9,A11,Th17;
    end;
  end;
A14: f,(sort_a f) are_fiberwise_equipotent by Def6;
A15: f,(sort_d f) are_fiberwise_equipotent by Def5;
A16: for j being Nat st j in dom (sort_a f) & (sort_a f).j=(
  sort_a f).1 holds 1<=j by FINSEQ_3:25;
  then
A17: (sort_a f).1=min (sort_a f) by A1,A8,A9,A10,Def2
    .=min f by A14,Th15;
A18: for j being Nat st j in dom (sort_d f) & (sort_d f).j=(
  sort_d f).1 holds 1<=j
  proof
    let j be Nat;
    assume that
A19: j in dom (sort_d f) and
     (sort_d f).j=(sort_d f).1;
    j in Seg len (sort_d f) by A19,FINSEQ_1:def 3;
    hence thesis by FINSEQ_1:1;
  end;
  then (sort_d f).1=max (sort_d f) by A1,A2,A3,A4,Def1
    .=max f by A15,Th14;
  hence thesis by A1,A2,A3,A4,A18,A8,A9,A10,A16,A17,Def1,Def2;
end;
