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
  x in dom f implies <*f*>..(1,x) = f.x & <*f,g*>..(1,x) = f.x &
    <*f,g,h*>..(1,x) = f.x
proof
A1: <*f,g,h*>.1 = f & 1 in Seg 1 by FINSEQ_1:2,TARSKI:def 1;
A2: dom <*f*> = Seg 1 & dom <*f,g*> = Seg 2 by FINSEQ_1:89;
A3: 1 in Seg 2 & 1 in Seg 3 by Th1,ENUMSET1:def 1,FINSEQ_1:2,TARSKI:def 2;
A4: dom <*f,g,h*> = Seg 3 by FINSEQ_1:89;
  <*f*>.1 = f & <*f,g*>.1 = f;
  hence thesis by A1,A3,A2,A4,FUNCT_5:38;
end;
