reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;
reserve s for Element of D*;

theorem
  for f being Function, x, y, z being set st x in dom f & y in dom f & z
  in dom f holds f*<*x,y,z*> = <*f.x,f.y,f.z*>
proof
  let f be Function;
  let x,y,z be set;
  assume that
A1: x in dom f and
A2: y in dom f & z in dom f;
  reconsider D = dom f, E = rng f as non empty set by A1,FUNCT_1:3;
  rng <*x,y,z*> = {x,y,z} by Lm2;
  then
A3: rng <*x,y,z*> = {x,y} \/ {z} by ENUMSET1:3;
  {x,y} c= D & {z} c= D by A1,A2,ZFMISC_1:31,32;
  then rng <*x,y,z*> c= D by A3,XBOOLE_1:8;
  then reconsider p = <*x,y,z*> as FinSequence of D by FINSEQ_1:def 4;
  reconsider g = f as Function of D,E by FUNCT_2:def 1,RELSET_1:4;
  thus f*<*x,y,z*> = g*p .= <*f.x,f.y,f.z*> by Th35;
end;
