reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem Th89:
  for C being CategorySet holds (C is U-set iff SetToCat C is U-element)
  proof
    let C be CategorySet;
A1: C = [C`1,C`2,C`3,C`4,C`5];
    hereby
      assume C is U-set;
      then
A2:   C`1 is Element of U & C`2 is Element of U & C`3 is Element of U &
        C`4 is Element of U & C`5 is Element of U by A1,Th14;
      ex y1,y2 be set,y3,y4 be Function of y2,y1,
      y5 be PartFunc of [:y2,y2:],y2 st
      y1 = C`1 & y2 = C`2 & y3 = C`3 & y4 = C`4 & y5 = C`5 &
      CatStr(# y1,y2,y3,y4,y5 #) is Category by Def27;
      hence SetToCat C is U-element by A2,Def31;
    end;
    assume A3: SetToCat C is U-element;
    set C2 = SetToCat C;
    consider y1,y2 be set,y3,y4 be Function of y2,y1,
    y5 be PartFunc of [:y2,y2:],y2 such that
A4: y1 = C`1 & y2 = C`2 & y3 = C`3 & y4 = C`4 & y5 = C`5 &
    SetToCat C = CatStr(# y1,y2,y3,y4,y5 #) by Def31;
    reconsider y1,y2 as Element of U by A4,A3;
    reconsider y22 = [:y2,y2:] as Element of U;
    y3 c= [:y2,y1:] & y4 c= [:y2,y1:] & y5 c= [:y22,y2:];
    then reconsider y3,y4,y5 as Element of U by CLASSES4:13;
    CatToSet C2 = [y1,y2,y3,y4,y5] by A4;
    then CatToSet SetToCat C is U-set;
    hence C is U-set by Th86;
  end;
