reserve k, l, m, n, i, j for Nat,
  K, N for non empty Subset of NAT,
  Ke, Ne, Me for Subset of NAT,
  X,Y for set;
reserve f for Function of Segm n,Segm k;
reserve x,y for set;

theorem Th62:
  for F be Function of Ne,Ke st F is 'increasing holds min* rng F= F.min* dom F
proof
  let F be Function of Ne,Ke such that
A1: F is 'increasing;
  now
    per cases;
    suppose
A2:   rng F is empty;
      then
A3:   min* rng F=0 by NAT_1:def 1;
      dom F={} by A2,RELAT_1:42;
      hence thesis by A3,FUNCT_1:def 2;
    end;
    suppose
A4:   rng F is non empty;
      then reconsider rngF=rng F,Ke as non empty Subset of NAT by XBOOLE_1:1;
      reconsider domF=dom F as non empty Subset of NAT by A4,FUNCT_2:def 1
,RELAT_1:42;
      set md=min* domF;
      set mr=min* rngF;
      mr=F.md
      proof
A5:     md in dom F by NAT_1:def 1;
        then F.md in rngF by FUNCT_1:def 3;
        then
A6:     mr <=F.md by NAT_1:def 1;
        assume mr<>F.md;
        then
A7:     mr < F.md by A6,XXREAL_0:1;
A8:     md in domF by NAT_1:def 1;
A9:     md in dom F by NAT_1:def 1;
        mr in rngF by NAT_1:def 1;
        then consider x being object such that
A10:    x in dom F and
A11:    F.x=mr by FUNCT_1:def 3;
A12:    F.md in {F.md} by TARSKI:def 1;
        F.x in {mr} by A11,TARSKI:def 1;
        then reconsider
        Fmr=F"{mr},Fmd=F"{F.md} as non empty Subset of NAT by A10,A12,A9,
FUNCT_1:def 7,XBOOLE_1:1;
A13:    mr in rngF by NAT_1:def 1;
        min* Fmr in Fmr by NAT_1:def 1;
        then min* Fmr in domF by FUNCT_1:def 7;
        then
A14:    min* Fmr >= md by NAT_1:def 1;
        F.md in {F.md} by TARSKI:def 1;
        then md in Fmd by A8,FUNCT_1:def 7;
        then
A15:    md >= min* Fmd by NAT_1:def 1;
        F.md in rng F by A5,FUNCT_1:def 3;
        then min* F"{mr}< min* F"{F.md} by A1,A7,A13;
        hence contradiction by A14,A15,XXREAL_0:2;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
