let IPP be 2-dimensional Desarguesian IncProjSp; :: thesis: for o being POINT of
for A being LINE of st not o on A holds
IncProj A,o,A = id (CHAIN A)
let o be POINT of ; :: thesis: for A being LINE of st not o on A holds
IncProj A,o,A = id (CHAIN A)
let A be LINE of ; :: thesis: ( not o on A implies IncProj A,o,A = id (CHAIN A) )
set f = IncProj A,o,A;
assume A1: not o on A ; :: thesis: IncProj A,o,A = id (CHAIN A)
A2: for x being set st x in CHAIN A holds
(IncProj A,o,A) . x = x

proof

let x be set ; :: thesis: ( x in CHAIN A implies (IncProj A,o,A) . x = x )
assume x in CHAIN A ; :: thesis: (IncProj A,o,A) . x = x
then ex x' being Element of the Points of IPP st
( x = x' & x' on A ) ;
hence (IncProj A,o,A) . x = x by A1, PROJRED1:27; :: thesis: verum

end;

dom (IncProj A,o,A) = CHAIN A by A1, Th5;
hence IncProj A,o,A = id (CHAIN A) by A2, FUNCT_1:34; :: thesis: verum