let r be Real; :: thesis: for i, j, n being Nat st 1 <= i & i < j & j <= n holds
Mx2Tran (Rotation (i,j,n,r)) is {i,j} -support-yielding
let i, j, n be Nat; :: thesis: ( 1 <= i & i < j & j <= n implies Mx2Tran (Rotation (i,j,n,r)) is {i,j} -support-yielding )
set M = Mx2Tran (Rotation (i,j,n,r));
assume A1: ( 1 <= i & i < j & j <= n ) ; :: thesis: Mx2Tran (Rotation (i,j,n,r)) is {i,j} -support-yielding
let f be Function; :: according to MATRTOP3:def 1 :: thesis: for x being set st f in dom (Mx2Tran (Rotation (i,j,n,r))) & ((Mx2Tran (Rotation (i,j,n,r))) . f) . x <> f . x holds
x in {i,j}
let x be set ; :: thesis: ( f in dom (Mx2Tran (Rotation (i,j,n,r))) & ((Mx2Tran (Rotation (i,j,n,r))) . f) . x <> f . x implies x in {i,j} )
assume that
A2: f in dom (Mx2Tran (Rotation (i,j,n,r))) and
A3: ((Mx2Tran (Rotation (i,j,n,r))) . f) . x <> f . x ; :: thesis: x in {i,j}
reconsider p = f as Point of (TOP-REAL n) by A2, FUNCT_2:52;
len p = n by CARD_1:def 7;
then A4: dom p = Seg n by FINSEQ_1:def 3;
len ((Mx2Tran (Rotation (i,j,n,r))) . p) = n by CARD_1:def 7;
then A5: dom ((Mx2Tran (Rotation (i,j,n,r))) . p) = 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,j}

then ((Mx2Tran (Rotation (i,j,n,r))) . p) . x = {} by A5, FUNCT_1:def 2;
hence x in {i,j} by A3, A4, A6, FUNCT_1:def 2; :: thesis: verum

end;

suppose A7: x in Seg n ; :: thesis: x in {i,j}

((Mx2Tran (Rotation (i,j,n,r))) . p) . x <> p . x by A3;
then ( x = i or x = j ) by A1, A7, Th20;
hence x in {i,j} by TARSKI:def 2; :: thesis: verum

end;

end;