
theorem
  for f being one-to-one Function for S1,S2 being Circuit-like non
empty ManySortedSign st the ResultSort of S1 c= f & the ResultSort of S2 c= f
  holds S1+*S2 is Circuit-like
proof
  let f be one-to-one Function;
  let S1,S2 be Circuit-like non empty ManySortedSign such that
A1: the ResultSort of S1 c= f and
A2: the ResultSort of S2 c= f;
  let S be non void non empty ManySortedSign;
  set r1 = the ResultSort of S1, r2 =the ResultSort of S2;
  set r = the ResultSort of S;
A3: r1+*r2 c= r1 \/ r2 by FUNCT_4:29;
  assume S = S1+*S2;
  then
A4: r = r1+*r2 by Def2;
  r1 \/ r2 c= f by A1,A2,XBOOLE_1:8;
  then
A5: r1+*r2 c= f by A3;
  then
A6: dom r c= dom f by A4,GRFUNC_1:2;
  let o1, o2 be Gate of S;
A7: dom r = the carrier' of S by FUNCT_2:def 1;
  then
A8: o1 in dom r;
A9: o2 in dom r by A7;
  assume
A10: the_result_sort_of o1 = the_result_sort_of o2;
A11: r.o2 = f.o2 by A4,A7,A5,GRFUNC_1:2;
  r.o1 = f.o1 by A4,A7,A5,GRFUNC_1:2;
  hence thesis by A10,A8,A9,A11,A6,FUNCT_1:def 4;
end;
