The co-hosts were ahead at half time in the opening game of Euro 2012 but a disappointing second half saw Dimitris Salpigidis equalise before Szczesny was sent off for a foul on the same player in the 69th minute.
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"We were disappointed after the draw yesterday but now we saw today the fans which are still motivated and enthusiastic," Szczesny told reporters.
"We are looking forward to the next game, we have respect for Russia but we know also what we are able to do."
Szczesny’s team-mate Lukasz Piszczek, who impressed many with his marauding runs down the right hand side, insisted that he was pleased with the point gained against a team 47 places above Poland in the FIFA world rankings.
"We are not only disappointed but also a little bit satisfied,” Piszczek told reporters.
"The draw was for both sides okay. The atmosphere was good today, we wanted the win but we must be satisfied with the draw."
Midfielder Dariusz Dudka, meanwhile, revealed that everyone in the Polish camp is now fully focused on their next match and that they had already started analysing their Russian opponents.
"The draw [against Greece] is not a bad result," Dudka opined. "We are still positive and we are looking forward to play against Russia.
"We have seen the Russia game and the Czechs were better in the first 15 minutes but Russia scored the goals. They have a good team but we can win."