set X = the non empty set ;
set F = the Function of [: the non empty set , the non empty set :],REAL;
set O = the Element of the non empty set ;
set B = the BinOp of the non empty set ;
set G = the Function of [:REAL, the non empty set :], the non empty set ;
take RLSMetrStruct(# the non empty set , the Function of [: the non empty set , the non empty set :],REAL, the Element of the non empty set , the BinOp of the non empty set , the Function of [:REAL, the non empty set :], the non empty set #) ; :: thesis: ( not RLSMetrStruct(# the non empty set , the Function of [: the non empty set , the non empty set :],REAL, the Element of the non empty set , the BinOp of the non empty set , the Function of [:REAL, the non empty set :], the non empty set #) is empty & RLSMetrStruct(# the non empty set , the Function of [: the non empty set , the non empty set :],REAL, the Element of the non empty set , the BinOp of the non empty set , the Function of [:REAL, the non empty set :], the non empty set #) is strict )
thus not the carrier of RLSMetrStruct(# the non empty set , the Function of [: the non empty set , the non empty set :],REAL, the Element of the non empty set , the BinOp of the non empty set , the Function of [:REAL, the non empty set :], the non empty set #) is empty ; :: according to STRUCT_0:def 1 :: thesis: RLSMetrStruct(# the non empty set , the Function of [: the non empty set , the non empty set :],REAL, the Element of the non empty set , the BinOp of the non empty set , the Function of [:REAL, the non empty set :], the non empty set #) is strict
thus RLSMetrStruct(# the non empty set , the Function of [: the non empty set , the non empty set :],REAL, the Element of the non empty set , the BinOp of the non empty set , the Function of [:REAL, the non empty set :], the non empty set #) is strict ; :: thesis: verum