reserve n,m for Nat;

theorem
  for f being real-valued FinSequence,r1,r2 being Real st f=<*r1,r2*> holds
  min f=min(r1,r2) & min_p f=IFEQ(r1,min(r1,r2),1,2)
proof
  let f be real-valued FinSequence,r1,r2 be Real;
  assume
A1: f=<*r1,r2*>;
  then
A2: len f=2 by FINSEQ_1:44;
  then
A3: f.1>=f.(min_p f) by Th2;
A4: f.2=r2 by A1;
A5: (min_p f) in dom f by A2,Def2;
  then
A6: 1<= (min_p f) by FINSEQ_3:25;
A7: f.2>=f.(min_p f) by A2,Th2;
A8: f.1=r1 by A1;
A9: (min_p f)<=len f by A5,FINSEQ_3:25;
  per cases;
  suppose
    r1<=r2;
    then
A10: min(r1,r2)>=min f by A8,A3,XXREAL_0:def 9;
    now
      per cases;
      case
        min_p f<len f;
        then min_p f <1+1 by A1,FINSEQ_1:44;
        then min_p f <=1 by NAT_1:13;
        then
A11:    min_p f=1 by A6,XXREAL_0:1;
        then min f >=min(r1,r2) by A8,XXREAL_0:17;
        then min f=min(r1,r2) by A10,XXREAL_0:1;
        hence thesis by A8,A11,FUNCOP_1:def 8;
      end;
      case
        min_p f>=len f;
        then
A12:    min_p f=2 by A2,A9,XXREAL_0:1;
        then min f >=min(r1,r2) by A4,XXREAL_0:17;
        then
A13:    min f=min(r1,r2) by A10,XXREAL_0:1;
        1 in dom f by A2,FINSEQ_3:25;
        then r1<>r2 by A2,A8,A4,A12,Def2;
        hence thesis by A4,A12,A13,FUNCOP_1:def 8;
      end;
    end;
    hence thesis;
  end;
  suppose
    r1>r2;
    then
A14: min(r1,r2)>=min f by A4,A7,XXREAL_0:def 9;
    now
      per cases;
      case
        min_p f<len f;
        then min_p f <1+1 by A1,FINSEQ_1:44;
        then min_p f <=1 by NAT_1:13;
        then
A15:    min_p f=1 by A6,XXREAL_0:1;
        then min f >=min(r1,r2) by A8,XXREAL_0:17;
        then min f=min(r1,r2) by A14,XXREAL_0:1;
        hence thesis by A8,A15,FUNCOP_1:def 8;
      end;
      case
        min_p f>=len f;
        then
A16:    min_p f=2 by A2,A9,XXREAL_0:1;
        then min f >=min(r1,r2) by A4,XXREAL_0:17;
        then
A17:    min f=min(r1,r2) by A14,XXREAL_0:1;
        1 in dom f by A2,FINSEQ_3:25;
        then r1<>r2 by A2,A8,A4,A16,Def2;
        hence thesis by A4,A16,A17,FUNCOP_1:def 8;
      end;
    end;
    hence thesis;
  end;
end;
