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 Th20:
  (minfuncreal(A)).(f,(maxfuncreal(A)).(g,h)) = (maxfuncreal(A)).(
  (minfuncreal(A)).(f,g),(minfuncreal(A)).(f,h))
proof
  now
    let x be Element of A;
A1: x in dom (minreal.:(f,g)) by Lm6;
A2: x in dom (minreal.:(f,h)) by Lm6;
A3: x in dom (minreal.:(f,(maxreal.:(g,h)))) by Lm6;
A4: x in dom (maxreal.:(minreal.:(f,g),minreal.:(f,h))) by Lm6;
A5: x in dom (maxreal.:(g,h)) by Lm6;
    thus (minfuncreal(A)).(f,(maxfuncreal(A)).(g,h)).x =((minfuncreal(A)).(f,
    maxreal.:(g,h))).x by Def4
      .=(minreal.:(f,maxreal.:(g,h))).x by Def5
      .=minreal.(f.x,(maxreal.:(g,h)).x) by A3,FUNCOP_1:22
      .=minreal.(f.x,(maxreal.(g.x,h.x))) by A5,FUNCOP_1:22
      .=maxreal.(minreal.(f.x,g.x),minreal.(f.x,h.x)) by Th7
      .=maxreal.(minreal.:(f,g).x,minreal.(f.x,h.x)) by A1,FUNCOP_1:22
      .=maxreal.(minreal.:(f,g).x,minreal.:(f,h).x) by A2,FUNCOP_1:22
      .=maxreal.:(minreal.:(f,g),minreal.:(f,h)).x by A4,FUNCOP_1:22
      .=(maxfuncreal(A)).(minreal.:(f,g),minreal.:(f,h)).x by Def4
      .=(maxfuncreal(A)).((minfuncreal(A)).(f,g),minreal.:(f,h)).x by Def5
      .=(maxfuncreal(A)).((minfuncreal(A)).(f,g),(minfuncreal(A)).(f,h)).x
    by Def5;
  end;
  hence thesis by FUNCT_2:63;
end;
