
theorem Th12:
  for C being non empty AltCatStr
  for a,b,f being object st f in (the Arrows of C).(a,b)
  ex o1,o2 being Object of C
  st o1 = a & o2 = b & f in <^o1,o2^> & f is Morphism of o1,o2
proof
  let C be non empty AltCatStr;
  let a,b,f be object;
  assume
A1: f in (the Arrows of C).(a,b);
  then [a,b] in dom the Arrows of C by FUNCT_1:def 2;
  then [a,b] in [:the carrier of C, the carrier of C:];
  then reconsider o1 = a, o2 = b as Object of C by ZFMISC_1:87;
  take o1, o2;
  thus thesis by A1,ALTCAT_1:def 1;
end;
