reserve x,y,y1,y2 for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,h,g,h1 for Membership_Func of C;

theorem Th17:
  max(f,UMF(C)) = UMF(C) & min(f,UMF(C)) = f & max(f,EMF(C)) = f &
  min(f,EMF(C)) = EMF(C)
proof
A1: C = dom max(f,EMF(C)) & C = dom min(f,EMF(C)) by FUNCT_2:def 1;
A2: for x being Element of C st x in C holds min(f,UMF(C)).x = f.x
  proof
    let x be Element of C;
A3: f.x <= (UMF(C)).x by Th15;
    min(f,UMF(C)).x = min(f.x,(UMF(C)).x) by Def3
      .=f.x by A3,XXREAL_0:def 9;
    hence thesis;
  end;
A4: for x being Element of C st x in C holds min(f,EMF(C)).x = (EMF(C)).x
  proof
    let x be Element of C;
A5: (EMF(C)).x <= f.x by Th15;
    min(f,EMF(C)).x = min(f.x,(EMF(C)).x) by Def3
      .=(EMF(C)).x by A5,XXREAL_0:def 9;
    hence thesis;
  end;
A6: for x being Element of C st x in C holds max(f,EMF(C)).x = f.x
  proof
    let x be Element of C;
A7: (EMF(C)).x <= f.x by Th15;
    max(f,EMF(C)).x = max(f.x,(EMF(C)).x) by Def4
      .=f.x by A7,XXREAL_0:def 10;
    hence thesis;
  end;
  max(f,UMF(C)) c= UMF(C) & UMF(C) c= max(f,UMF(C)) by Th14,Th16;
  hence max(f,UMF(C)) = UMF(C) by Lm1;
A8: C = dom EMF(C) by FUNCT_2:def 1;
  C = dom min(f,UMF(C)) & C = dom f by FUNCT_2:def 1;
  hence thesis by A2,A6,A1,A8,A4,PARTFUN1:5;
end;
