8 !capm P@P4h@ AMFR p|'AF0 F`E,EEF0E$FFFCC AF p|'BFHHEF0FCC  p|'R0(>ptPtFtFt@td,.+w0 #+CCF @6^H p|'3KEVKK` mory.eKK#&JwxI{=h'P]^We proceed to show that harmonic sets remain harmonic after any unmber of perspectives. As a first step we shall prove Theorem: Any section of a harmonic set of lines is a harmonic set of points, and a harmonic set of points is projected from any point by a harmonic set of lines. Remark: This theorem is in two dual parts, and thus it will suffice to prove the latter part: If A,B,C,D are joined to a point P (outside their line) by lines a,b,c,d and if H(AB,CD), then H(ab,cd). Proof: Let P be used as a vertex of the triangle PQR in constructing D from A,B,C. Then the quadrilateral AD,DS,SQ,QA has two opposite vertices on AS=a, two others on BR=b, one vertex Q on c, and one vertex D on d. Hence H(ab,cd) Combining the above, we conclude that Theorem: Perspectivities preserve the harmonic relation: If ABCD =/\= A'B'C'D' and H(AB,CD), then H(A'B',C'D') Our definition for harmonic conjugacy involves A and B symmetrically and likewise C and D. Hence Theorem: H(AB,CD), H(AB,DC), H(BA,DC), H(BA,CD) are all equivalent.e their line) by lines a,b,c,d and if H(AB,CD), then H(ab,cd). Proof: K@I(< !)  p|'  p1H   XahLee.org, 2004.   0 @   1@F'`< p|'j~`$~`~`~`~` ~ ~!`~"`~%`~$`~-`~.`nCCCC?2' p|'k HT ?<Hh/ FFIFFCCFCC?g' p|'b0( p|' D0(>@tPtFtFt@td,.+w0 #+CC '  p|'m S=|?<nB'F F!PIF!PFCCFCC? @LF  p|'ae IuEE E EECCCC?  ? p|' S0Gt@tDth@t@Dd+w0Nt+CTVCB u%R' p|'o H(B P;h~OF%F(@IF(@FCCCTVCB? 2' p|'a0(ptPtFtFt@td,.+w0 #+CƨCe$  F' p|'P0(>ptPtFtFt@td,.+w0 #+CrC+Z B' p|'c0(@tPtFtFt@td,.+w0 #+DdC