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awk '/EFG/{foundEFG=1;t=""}
/LMN/&&foundEFG{print;next}
/LMN/{t=t?t RS $0:$0}
END{if(!foundEFG) print t}' file

awk '/EFG/{p=1} # set variable p to 1 when the pattern EFG is found
/LMN/&&p{ # when pattern LMN is found AND p is set (when EFG has been found)
print "LMN at line "NR} #...

awk '/EFG/{p=1}/LMN/&&p{print "LMN at line "NR}' file