let FdSp be FanodesSp; :: thesis: for a, b, c being Element of FdSp ex d being Element of FdSp st a,b congr c,d
let a, b, c be Element of FdSp; :: thesis: ex d being Element of FdSp st a,b congr c,d

A1: now

assume a = b ; :: thesis: ex d being Element of FdSp st a,b congr c,d
then a,b congr c,c by Def4;
hence ex d being Element of FdSp st a,b congr c,d ; :: thesis: verum

end;

A2: now

assume that
A3: a <> b and
A4: a,b,c is_collinear ; :: thesis: ex d being Element of FdSp st a,b congr c,d
consider p, q being Element of FdSp such that
A5: parallelogram a,b,p,q by A3, Th51;
not p,q,c is_collinear by A4, A5, Th37;
then consider d being Element of FdSp such that
A6: parallelogram p,q,c,d by Th42;
parallelogram p,q,a,b by A5, Th41;
then a,b congr c,d by A6, Def4;
hence ex d being Element of FdSp st a,b congr c,d ; :: thesis: verum

end;

now

assume that
a <> b and
A7: not a,b,c is_collinear ; :: thesis: ex d being Element of FdSp st a,b congr c,d
ex d being Element of FdSp st parallelogram a,b,c,d by A7, Th42;
hence ex d being Element of FdSp st a,b congr c,d by Th61; :: thesis: verum

end;

hence ex d being Element of FdSp st a,b congr c,d by A1, A2; :: thesis: verum