reserve x,y for Real;
reserve a,b,c for Element of Real_Lattice;
reserve p,q,r for Element of Real_Lattice;
reserve A,B for non empty set;
reserve f,g,h for Element of Funcs(A,REAL);

theorem Th12:
  (maxfuncreal(A)).(f,(minfuncreal(A)).(f,g))=f
proof
  now
    let x be Element of A;
A1: x in dom (minreal.:(f,g)) by Lm6;
A2: x in dom (maxreal.:(f,(minreal.:(f,g)))) by Lm6;
    thus (maxfuncreal(A)).(f,(minfuncreal(A)).(f,g)).x =((maxfuncreal(A)).(f,
    minreal.:(f,g))).x by Def5
      .=(maxreal.:(f,minreal.:(f,g))).x by Def4
      .=maxreal.(f.x,(minreal.:(f,g)).x) by A2,FUNCOP_1:22
      .=maxreal.(f.x,(minreal.(f.x,g.x))) by A1,FUNCOP_1:22
      .=f.x by Th5;
  end;
  hence thesis by FUNCT_2:63;
end;
