let OAS be OAffinSpace; :: thesis: for f being Permutation of the carrier of OAS st f is dilatation holds
ex f9 being Permutation of the carrier of (Lambda OAS) st
( f = f9 & f9 is_DIL_of Lambda OAS )
let f be Permutation of the carrier of OAS; :: thesis: ( f is dilatation implies ex f9 being Permutation of the carrier of (Lambda OAS) st
( f = f9 & f9 is_DIL_of Lambda OAS ) )
A1: Lambda OAS = AffinStruct(# the carrier of OAS,(lambda the CONGR of OAS) #) by DIRAF:def 2;
then reconsider f9 = f as Permutation of the carrier of (Lambda OAS) ;
assume f is dilatation ; :: thesis: ex f9 being Permutation of the carrier of (Lambda OAS) st
( f = f9 & f9 is_DIL_of Lambda OAS )
then A2: f is_FormalIz_of lambda the CONGR of OAS ;
take f9 ; :: thesis: ( f = f9 & f9 is_DIL_of Lambda OAS )
thus ( f = f9 & f9 is_DIL_of Lambda OAS ) by A2, A1; :: thesis: verum