let x be object ; :: according to VALUED_2:def 28 :: thesis: ( not x in dom (RealFuncExtMult A) or (RealFuncExtMult A) . x is set )
set IT = RealFuncExtMult A;
assume x in dom (RealFuncExtMult A) ; :: thesis: (RealFuncExtMult A) . x is set
then (RealFuncExtMult A) . x is Function of A,REAL by FUNCT_2:66, PARTFUN1:4;
hence (RealFuncExtMult A) . x is real-valued Function ; :: thesis: verum