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;
