let i, n be Nat; :: thesis: ( i in Seg n implies Mx2Tran (AxialSymmetry (i,n)) is {i} -support-yielding )
set M = Mx2Tran (AxialSymmetry (i,n));
assume A1: i in Seg n ; :: thesis: Mx2Tran (AxialSymmetry (i,n)) is {i} -support-yielding
let f be Function; :: according to MATRTOP3:def 1 :: thesis: for x being set st f in dom (Mx2Tran (AxialSymmetry (i,n))) & ((Mx2Tran (AxialSymmetry (i,n))) . f) . x <> f . x holds
x in {i}
let x be set ; :: thesis: ( f in dom (Mx2Tran (AxialSymmetry (i,n))) & ((Mx2Tran (AxialSymmetry (i,n))) . f) . x <> f . x implies x in {i} )
assume f in dom (Mx2Tran (AxialSymmetry (i,n))) ; :: thesis: ( not ((Mx2Tran (AxialSymmetry (i,n))) . f) . x <> f . x or x in {i} )
then reconsider F = f as Point of (TOP-REAL n) by FUNCT_2:52;
assume A2: ((Mx2Tran (AxialSymmetry (i,n))) . f) . x <> f . x ; :: thesis: x in {i}
len ((Mx2Tran (AxialSymmetry (i,n))) . F) = n by CARD_1:def 7;
then A3: dom ((Mx2Tran (AxialSymmetry (i,n))) . F) = Seg n by FINSEQ_1:def 3;
A4: len F = n by CARD_1:def 7;
then A5: dom F = Seg n by FINSEQ_1:def 3;

per cases ( not x in Seg n or x in Seg n ) ;

suppose A6: not x in Seg n ; :: thesis: x in {i}

then ((Mx2Tran (AxialSymmetry (i,n))) . F) . x = {} by A3, FUNCT_1:def 2
.= F . x by A5, A6, FUNCT_1:def 2 ;
hence x in {i} by A2; :: thesis: verum

end;

suppose x in Seg n ; :: thesis: x in {i}

then x = i by A1, A2, A4, Th8;
hence x in {i} by TARSKI:def 1; :: thesis: verum

end;

end;