let X be ComplexNormSpace; :: thesis: for f being Element of BoundedLinearOperators (X,X) holds
( f * (FuncUnit X) = f & (FuncUnit X) * f = f )
let f be Element of BoundedLinearOperators (X,X); :: thesis: ( f * (FuncUnit X) = f & (FuncUnit X) * f = f )
id the carrier of X is Lipschitzian LinearOperator of X,X by Th3;
then id the carrier of X is Element of BoundedLinearOperators (X,X) by CLOPBAN1:def 7;
then A1: modetrans ((id the carrier of X),X,X) = id the carrier of X by CLOPBAN1:def 9;
hence f * (FuncUnit X) = modetrans (f,X,X) by Th6
.= f by CLOPBAN1:def 9 ;
:: thesis: (FuncUnit X) * f = f
thus (FuncUnit X) * f = modetrans (f,X,X) by A1, Th6
.= f by CLOPBAN1:def 9 ; :: thesis: verum