theorem Th82:
  opposite(a,o)=opposite(b,o) implies a=b
proof
  assume
A1: opposite(a,o)=opposite(b,o);
  congr a,o,o,opposite(a,o) by Th81;
  then
A2: congr opposite(a,o),o,o,a by Th69;
  congr b,o,o,opposite(b,o) by Th81;
  then congr opposite(a,o),o,o,b by A1,Th69;
  hence thesis by A2,Th62;
end;
