reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;
reserve J for Nat;
reserve n for Nat;
reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem Th140:
  dom <:<*f1,f2*>:> = dom f1 /\ dom f2 &
    for x st x in dom f1 /\ dom f2 holds
      <:<*f1,f2*>:>.x = <*f1.x,f2.x*>
proof
A1: dom <*f1,f2*> = Seg 2 by FINSEQ_1:89;
A2: rng <:<*f1,f2*>:> c= product rngs <*f1,f2*> & rngs <*f1,f2*> = <*rng f1,
  rng f2*> by Th131,FUNCT_6:29;
  thus
A3: dom <:<*f1,f2*>:> = meet doms <*f1,f2*> by FUNCT_6:29
    .= meet <*dom f1, dom f2*> by Th131
    .= dom f1 /\ dom f2 by Th134;
  let x;
  assume
A4: x in dom f1 /\ dom f2;
  then <:<*f1,f2*>:>.x in rng <:<*f1,f2*>:> by A3,FUNCT_1:def 3;
  then consider g such that
A5: <:<*f1,f2*>:>.x = g and
A6: dom g = dom <*rng f1, rng f2*> and
  for y being object st y in dom <*rng f1, rng f2*>
holds g.y in <*rng f1, rng f2*> .y
  by A2,CARD_3:def 5;
A7: dom g = Seg 2 by A6,FINSEQ_1:89;
A8: 1 in Seg 2 by FINSEQ_1:2,TARSKI:def 2;
  reconsider g as FinSequence by A7,FINSEQ_1:def 2;
A9: 2 in Seg 2 by FINSEQ_1:2,TARSKI:def 2;
  then
A10: g.2 = (uncurry <*f1,f2*>).(2,x) by A3,A4,A5,A7,FUNCT_6:31;
  <*f1,f2*>.2 = f2 & x in dom f2 by A4,XBOOLE_0:def 4;
  then
A11: (uncurry <*f1,f2*>).(2,x) = f2.x by A1,A9,FUNCT_5:38;
A12: len g = 2 by A7,FINSEQ_1:def 3;
  <*f1,f2*>.1 = f1 & x in dom f1 by A4,XBOOLE_0:def 4;
  then
A13: (uncurry <*f1,f2*>).(1,x) = f1.x by A1,A8,FUNCT_5:38;
  g.1 = (uncurry <*f1,f2*>).(1,x) by A3,A4,A5,A7,A8,FUNCT_6:31;
  hence thesis by A5,A13,A10,A11,A12,FINSEQ_1:44;
end;
