reserve C1, C2 for Coherence_Space;

theorem Th97:
  for C1,C2 being Coherence_Space, a being Element of C1 [*] C2
  holds proj1 a in C1 & proj2 a in C2 & a c= [:proj1 a, proj2 a:]
proof
  let C1,C2 be Coherence_Space, a be Element of C1 [*] C2;
  consider a1 being Element of C1, a2 being Element of C2 such that
A1: a c= [:a1,a2:] by Th96;
  proj1 a c= a1 & proj2 a c= a2 by A1,FUNCT_5:11;
  hence proj1 a in C1 & proj2 a in C2 by CLASSES1:def 1;
  let x be object;
  assume
A2: x in a;
  then
A3: x = [x`1,x`2] by A1,MCART_1:21;
  then x`1 in proj1 a & x`2 in proj2 a by A2,XTUPLE_0:def 12,def 13;
  hence thesis by A3,ZFMISC_1:87;
end;
