theorem Th17:
  j in dom(f|i) & j+1 in dom(f|i) implies LSeg(f,j) = LSeg(f|i,j)
proof
  assume that
A1: j in dom(f|i) and
A2: j+1 in dom(f|i);
A3: 1 <= j & j+1 <= len(f|i) by A1,A2,FINSEQ_3:25;
  set p1 = (f|i)/.j, p2 = (f|i)/.(j+1);
A4: f|i = f| (Seg i) by FINSEQ_1:def 16;
  then j in dom f /\ (Seg i) by A1,RELAT_1:61;
  then j in dom f by XBOOLE_0:def 4;
  then
A5: 1 <= j by FINSEQ_3:25;
  j+1 in dom f /\ (Seg i) by A2,A4,RELAT_1:61;
  then j+1 in dom f by XBOOLE_0:def 4;
  then
A6: j+1 <= len f by FINSEQ_3:25;
  p1 = f/.j & p2 = f/.(j+1) by A1,A2,FINSEQ_4:70;
  then LSeg(f,j) = LSeg(p1,p2) by A5,A6,TOPREAL1:def 3;
  hence thesis by A3,TOPREAL1:def 3;
end;
